Properties

Label 14-115e7-1.1-c5e7-0-0
Degree $14$
Conductor $2.660\times 10^{14}$
Sign $1$
Analytic cond. $7.26120\times 10^{8}$
Root an. cond. $4.29466$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 3·3-s − 15·4-s − 175·5-s − 12·6-s + 33·7-s + 58·8-s − 626·9-s − 700·10-s + 1.37e3·11-s + 45·12-s + 605·13-s + 132·14-s + 525·15-s + 167·16-s + 2.50e3·17-s − 2.50e3·18-s − 115·19-s + 2.62e3·20-s − 99·21-s + 5.49e3·22-s + 3.70e3·23-s − 174·24-s + 1.75e4·25-s + 2.42e3·26-s − 1.96e3·27-s − 495·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.192·3-s − 0.468·4-s − 3.13·5-s − 0.136·6-s + 0.254·7-s + 0.320·8-s − 2.57·9-s − 2.21·10-s + 3.42·11-s + 0.0902·12-s + 0.992·13-s + 0.179·14-s + 0.602·15-s + 0.163·16-s + 2.10·17-s − 1.82·18-s − 0.0730·19-s + 1.46·20-s − 0.0489·21-s + 2.41·22-s + 1.45·23-s − 0.0616·24-s + 28/5·25-s + 0.702·26-s − 0.517·27-s − 0.119·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 23^{7}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(5^{7} \cdot 23^{7}\)
Sign: $1$
Analytic conductor: \(7.26120\times 10^{8}\)
Root analytic conductor: \(4.29466\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((14,\ 5^{7} \cdot 23^{7} ,\ ( \ : [5/2]^{7} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(12.15937164\)
\(L(\frac12)\) \(\approx\) \(12.15937164\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 + p^{2} T )^{7} \)
23 \( ( 1 - p^{2} T )^{7} \)
good2 \( 1 - p^{2} T + 31 T^{2} - 121 p T^{3} + 749 p T^{4} - 215 p^{5} T^{5} + 1995 p^{5} T^{6} - 18625 p^{4} T^{7} + 1995 p^{10} T^{8} - 215 p^{15} T^{9} + 749 p^{16} T^{10} - 121 p^{21} T^{11} + 31 p^{25} T^{12} - p^{32} T^{13} + p^{35} T^{14} \)
3 \( 1 + p T + 635 T^{2} + 1915 p T^{3} + 7762 p^{3} T^{4} + 154031 p^{3} T^{5} + 188300 p^{5} T^{6} + 5585498 p^{5} T^{7} + 188300 p^{10} T^{8} + 154031 p^{13} T^{9} + 7762 p^{18} T^{10} + 1915 p^{21} T^{11} + 635 p^{25} T^{12} + p^{31} T^{13} + p^{35} T^{14} \)
7 \( 1 - 33 T + 86173 T^{2} - 98167 p T^{3} + 3381810942 T^{4} + 25166217879 T^{5} + 11724343503454 p T^{6} + 932556634234166 T^{7} + 11724343503454 p^{6} T^{8} + 25166217879 p^{10} T^{9} + 3381810942 p^{15} T^{10} - 98167 p^{21} T^{11} + 86173 p^{25} T^{12} - 33 p^{30} T^{13} + p^{35} T^{14} \)
11 \( 1 - 1373 T + 1488600 T^{2} - 1170972260 T^{3} + 777859969498 T^{4} - 434724092454699 T^{5} + 213641162218760103 T^{6} - 90872758608138294520 T^{7} + 213641162218760103 p^{5} T^{8} - 434724092454699 p^{10} T^{9} + 777859969498 p^{15} T^{10} - 1170972260 p^{20} T^{11} + 1488600 p^{25} T^{12} - 1373 p^{30} T^{13} + p^{35} T^{14} \)
13 \( 1 - 605 T + 1281743 T^{2} - 355494995 T^{3} + 581155930252 T^{4} + 101985007072035 T^{5} + 7881585685234776 p T^{6} + 660120254293964538 p^{2} T^{7} + 7881585685234776 p^{6} T^{8} + 101985007072035 p^{10} T^{9} + 581155930252 p^{15} T^{10} - 355494995 p^{20} T^{11} + 1281743 p^{25} T^{12} - 605 p^{30} T^{13} + p^{35} T^{14} \)
17 \( 1 - 2505 T + 8946853 T^{2} - 14709443433 T^{3} + 29529335509906 T^{4} - 36250291844053321 T^{5} + 55448727845330797644 T^{6} - \)\(57\!\cdots\!10\)\( T^{7} + 55448727845330797644 p^{5} T^{8} - 36250291844053321 p^{10} T^{9} + 29529335509906 p^{15} T^{10} - 14709443433 p^{20} T^{11} + 8946853 p^{25} T^{12} - 2505 p^{30} T^{13} + p^{35} T^{14} \)
19 \( 1 + 115 T + 5867678 T^{2} - 1695831428 T^{3} + 17139954072668 T^{4} - 20750511151798507 T^{5} + 36718193662591686115 T^{6} - \)\(77\!\cdots\!20\)\( T^{7} + 36718193662591686115 p^{5} T^{8} - 20750511151798507 p^{10} T^{9} + 17139954072668 p^{15} T^{10} - 1695831428 p^{20} T^{11} + 5867678 p^{25} T^{12} + 115 p^{30} T^{13} + p^{35} T^{14} \)
29 \( 1 - 2440 T + 99272757 T^{2} - 252387653860 T^{3} + 4598462425629220 T^{4} - 11799813378273509716 T^{5} + \)\(13\!\cdots\!46\)\( T^{6} - \)\(31\!\cdots\!68\)\( T^{7} + \)\(13\!\cdots\!46\)\( p^{5} T^{8} - 11799813378273509716 p^{10} T^{9} + 4598462425629220 p^{15} T^{10} - 252387653860 p^{20} T^{11} + 99272757 p^{25} T^{12} - 2440 p^{30} T^{13} + p^{35} T^{14} \)
31 \( 1 - 13565 T + 233099548 T^{2} - 2117019487166 T^{3} + 21084300361356499 T^{4} - \)\(14\!\cdots\!94\)\( T^{5} + \)\(10\!\cdots\!39\)\( T^{6} - \)\(53\!\cdots\!06\)\( T^{7} + \)\(10\!\cdots\!39\)\( p^{5} T^{8} - \)\(14\!\cdots\!94\)\( p^{10} T^{9} + 21084300361356499 p^{15} T^{10} - 2117019487166 p^{20} T^{11} + 233099548 p^{25} T^{12} - 13565 p^{30} T^{13} + p^{35} T^{14} \)
37 \( 1 - 9414 T + 401400724 T^{2} - 3415108043804 T^{3} + 74933367487809630 T^{4} - \)\(54\!\cdots\!38\)\( T^{5} + \)\(82\!\cdots\!13\)\( T^{6} - \)\(48\!\cdots\!00\)\( T^{7} + \)\(82\!\cdots\!13\)\( p^{5} T^{8} - \)\(54\!\cdots\!38\)\( p^{10} T^{9} + 74933367487809630 p^{15} T^{10} - 3415108043804 p^{20} T^{11} + 401400724 p^{25} T^{12} - 9414 p^{30} T^{13} + p^{35} T^{14} \)
41 \( 1 - 13725 T + 527992042 T^{2} - 5649603381438 T^{3} + 138493427835512455 T^{4} - \)\(12\!\cdots\!12\)\( T^{5} + \)\(23\!\cdots\!95\)\( T^{6} - \)\(17\!\cdots\!08\)\( T^{7} + \)\(23\!\cdots\!95\)\( p^{5} T^{8} - \)\(12\!\cdots\!12\)\( p^{10} T^{9} + 138493427835512455 p^{15} T^{10} - 5649603381438 p^{20} T^{11} + 527992042 p^{25} T^{12} - 13725 p^{30} T^{13} + p^{35} T^{14} \)
43 \( 1 - 76694 T + 3269035833 T^{2} - 97056708237596 T^{3} + 2221786921979400593 T^{4} - \)\(41\!\cdots\!18\)\( T^{5} + \)\(63\!\cdots\!61\)\( T^{6} - \)\(83\!\cdots\!56\)\( T^{7} + \)\(63\!\cdots\!61\)\( p^{5} T^{8} - \)\(41\!\cdots\!18\)\( p^{10} T^{9} + 2221786921979400593 p^{15} T^{10} - 97056708237596 p^{20} T^{11} + 3269035833 p^{25} T^{12} - 76694 p^{30} T^{13} + p^{35} T^{14} \)
47 \( 1 - 59692 T + 2622134766 T^{2} - 78916726889118 T^{3} + 2000037432406826632 T^{4} - \)\(41\!\cdots\!12\)\( T^{5} + \)\(75\!\cdots\!83\)\( T^{6} - \)\(12\!\cdots\!76\)\( T^{7} + \)\(75\!\cdots\!83\)\( p^{5} T^{8} - \)\(41\!\cdots\!12\)\( p^{10} T^{9} + 2000037432406826632 p^{15} T^{10} - 78916726889118 p^{20} T^{11} + 2622134766 p^{25} T^{12} - 59692 p^{30} T^{13} + p^{35} T^{14} \)
53 \( 1 - 49536 T + 3334102848 T^{2} - 112948128058794 T^{3} + 4378048883877492106 T^{4} - \)\(11\!\cdots\!04\)\( T^{5} + \)\(30\!\cdots\!41\)\( T^{6} - \)\(61\!\cdots\!24\)\( T^{7} + \)\(30\!\cdots\!41\)\( p^{5} T^{8} - \)\(11\!\cdots\!04\)\( p^{10} T^{9} + 4378048883877492106 p^{15} T^{10} - 112948128058794 p^{20} T^{11} + 3334102848 p^{25} T^{12} - 49536 p^{30} T^{13} + p^{35} T^{14} \)
59 \( 1 - 44536 T + 5531075430 T^{2} - 189674475817466 T^{3} + 12561372181584275856 T^{4} - \)\(33\!\cdots\!76\)\( T^{5} + \)\(15\!\cdots\!99\)\( T^{6} - \)\(32\!\cdots\!32\)\( T^{7} + \)\(15\!\cdots\!99\)\( p^{5} T^{8} - \)\(33\!\cdots\!76\)\( p^{10} T^{9} + 12561372181584275856 p^{15} T^{10} - 189674475817466 p^{20} T^{11} + 5531075430 p^{25} T^{12} - 44536 p^{30} T^{13} + p^{35} T^{14} \)
61 \( 1 + 49097 T + 4653658864 T^{2} + 159693951107374 T^{3} + 9472061290090909370 T^{4} + \)\(26\!\cdots\!79\)\( T^{5} + \)\(12\!\cdots\!89\)\( T^{6} + \)\(27\!\cdots\!52\)\( T^{7} + \)\(12\!\cdots\!89\)\( p^{5} T^{8} + \)\(26\!\cdots\!79\)\( p^{10} T^{9} + 9472061290090909370 p^{15} T^{10} + 159693951107374 p^{20} T^{11} + 4653658864 p^{25} T^{12} + 49097 p^{30} T^{13} + p^{35} T^{14} \)
67 \( 1 - 788 T + 5086360246 T^{2} + 13634420433142 T^{3} + 10802079109165850608 T^{4} + \)\(83\!\cdots\!64\)\( T^{5} + \)\(14\!\cdots\!39\)\( T^{6} + \)\(17\!\cdots\!92\)\( T^{7} + \)\(14\!\cdots\!39\)\( p^{5} T^{8} + \)\(83\!\cdots\!64\)\( p^{10} T^{9} + 10802079109165850608 p^{15} T^{10} + 13634420433142 p^{20} T^{11} + 5086360246 p^{25} T^{12} - 788 p^{30} T^{13} + p^{35} T^{14} \)
71 \( 1 - 49521 T + 8477048438 T^{2} - 403793573318654 T^{3} + 35463830905398999921 T^{4} - \)\(15\!\cdots\!42\)\( T^{5} + \)\(93\!\cdots\!65\)\( T^{6} - \)\(34\!\cdots\!18\)\( T^{7} + \)\(93\!\cdots\!65\)\( p^{5} T^{8} - \)\(15\!\cdots\!42\)\( p^{10} T^{9} + 35463830905398999921 p^{15} T^{10} - 403793573318654 p^{20} T^{11} + 8477048438 p^{25} T^{12} - 49521 p^{30} T^{13} + p^{35} T^{14} \)
73 \( 1 + 3760 T + 5907040138 T^{2} + 85629815801500 T^{3} + 21272909098679047328 T^{4} + \)\(34\!\cdots\!48\)\( T^{5} + \)\(60\!\cdots\!13\)\( T^{6} + \)\(85\!\cdots\!84\)\( T^{7} + \)\(60\!\cdots\!13\)\( p^{5} T^{8} + \)\(34\!\cdots\!48\)\( p^{10} T^{9} + 21272909098679047328 p^{15} T^{10} + 85629815801500 p^{20} T^{11} + 5907040138 p^{25} T^{12} + 3760 p^{30} T^{13} + p^{35} T^{14} \)
79 \( 1 - 918 T + 12884174669 T^{2} + 183144373803044 T^{3} + 83465643034580975361 T^{4} + \)\(15\!\cdots\!54\)\( T^{5} + \)\(37\!\cdots\!05\)\( T^{6} + \)\(60\!\cdots\!04\)\( T^{7} + \)\(37\!\cdots\!05\)\( p^{5} T^{8} + \)\(15\!\cdots\!54\)\( p^{10} T^{9} + 83465643034580975361 p^{15} T^{10} + 183144373803044 p^{20} T^{11} + 12884174669 p^{25} T^{12} - 918 p^{30} T^{13} + p^{35} T^{14} \)
83 \( 1 - 99202 T + 26933210106 T^{2} - 1986531192435580 T^{3} + \)\(30\!\cdots\!64\)\( T^{4} - \)\(17\!\cdots\!10\)\( T^{5} + \)\(19\!\cdots\!35\)\( T^{6} - \)\(88\!\cdots\!36\)\( T^{7} + \)\(19\!\cdots\!35\)\( p^{5} T^{8} - \)\(17\!\cdots\!10\)\( p^{10} T^{9} + \)\(30\!\cdots\!64\)\( p^{15} T^{10} - 1986531192435580 p^{20} T^{11} + 26933210106 p^{25} T^{12} - 99202 p^{30} T^{13} + p^{35} T^{14} \)
89 \( 1 + 141676 T + 19585447479 T^{2} + 1716007279127984 T^{3} + \)\(18\!\cdots\!65\)\( T^{4} + \)\(14\!\cdots\!28\)\( T^{5} + \)\(13\!\cdots\!71\)\( T^{6} + \)\(92\!\cdots\!04\)\( T^{7} + \)\(13\!\cdots\!71\)\( p^{5} T^{8} + \)\(14\!\cdots\!28\)\( p^{10} T^{9} + \)\(18\!\cdots\!65\)\( p^{15} T^{10} + 1716007279127984 p^{20} T^{11} + 19585447479 p^{25} T^{12} + 141676 p^{30} T^{13} + p^{35} T^{14} \)
97 \( 1 - 28731 T + 41283749088 T^{2} - 1372891375766714 T^{3} + \)\(83\!\cdots\!38\)\( T^{4} - \)\(26\!\cdots\!21\)\( T^{5} + \)\(10\!\cdots\!41\)\( T^{6} - \)\(29\!\cdots\!04\)\( T^{7} + \)\(10\!\cdots\!41\)\( p^{5} T^{8} - \)\(26\!\cdots\!21\)\( p^{10} T^{9} + \)\(83\!\cdots\!38\)\( p^{15} T^{10} - 1372891375766714 p^{20} T^{11} + 41283749088 p^{25} T^{12} - 28731 p^{30} T^{13} + p^{35} T^{14} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.78308965781526124347554799190, −5.69925717033744538983798306240, −5.45459681581859950234384551806, −5.38224505609168475990684049413, −4.77480031794884297856178863791, −4.63368121290443820077766871231, −4.52642494444267474232847826211, −4.48114204936046289456726908743, −4.19931995312552446048122796019, −4.04845556795245561602304093384, −3.94969256054595103028631608388, −3.65197476062493453990025237941, −3.45006378026906794262316038185, −3.27615950209065676843795693015, −3.04917687630478499731679770384, −2.97620921916898031187536839681, −2.42651386993389425771808744816, −2.36840226193348725904225118056, −2.01247695319561389550726348062, −1.18288539991737129115002278449, −1.06270293561828920338506828092, −0.957774225339019551451002904291, −0.78682218372385581998508735186, −0.67449044564973493879964906853, −0.35912705523634177621590274227, 0.35912705523634177621590274227, 0.67449044564973493879964906853, 0.78682218372385581998508735186, 0.957774225339019551451002904291, 1.06270293561828920338506828092, 1.18288539991737129115002278449, 2.01247695319561389550726348062, 2.36840226193348725904225118056, 2.42651386993389425771808744816, 2.97620921916898031187536839681, 3.04917687630478499731679770384, 3.27615950209065676843795693015, 3.45006378026906794262316038185, 3.65197476062493453990025237941, 3.94969256054595103028631608388, 4.04845556795245561602304093384, 4.19931995312552446048122796019, 4.48114204936046289456726908743, 4.52642494444267474232847826211, 4.63368121290443820077766871231, 4.77480031794884297856178863791, 5.38224505609168475990684049413, 5.45459681581859950234384551806, 5.69925717033744538983798306240, 5.78308965781526124347554799190

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.