Properties

Label 14-177e7-1.1-c3e7-0-0
Degree $14$
Conductor $5.443\times 10^{15}$
Sign $-1$
Analytic cond. $1.35480\times 10^{7}$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $7$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 21·3-s − 15·4-s − 2·5-s − 59·7-s − 7·8-s + 252·9-s − 5·11-s + 315·12-s − 67·13-s + 42·15-s + 37·16-s − 23·17-s − 176·19-s + 30·20-s + 1.23e3·21-s − 218·23-s + 147·24-s − 527·25-s − 2.26e3·27-s + 885·28-s + 168·29-s − 604·31-s + 49·32-s + 105·33-s + 118·35-s − 3.78e3·36-s − 505·37-s + ⋯
L(s)  = 1  − 4.04·3-s − 1.87·4-s − 0.178·5-s − 3.18·7-s − 0.309·8-s + 28/3·9-s − 0.137·11-s + 7.57·12-s − 1.42·13-s + 0.722·15-s + 0.578·16-s − 0.328·17-s − 2.12·19-s + 0.335·20-s + 12.8·21-s − 1.97·23-s + 1.25·24-s − 4.21·25-s − 16.1·27-s + 5.97·28-s + 1.07·29-s − 3.49·31-s + 0.270·32-s + 0.553·33-s + 0.569·35-s − 17.5·36-s − 2.24·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 59^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 59^{7}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(3^{7} \cdot 59^{7}\)
Sign: $-1$
Analytic conductor: \(1.35480\times 10^{7}\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{177} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(7\)
Selberg data: \((14,\ 3^{7} \cdot 59^{7} ,\ ( \ : [3/2]^{7} ),\ -1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p T )^{7} \)
59 \( ( 1 - p T )^{7} \)
good2 \( 1 + 15 T^{2} + 7 T^{3} + 47 p^{2} T^{4} + 161 T^{5} + 195 p^{3} T^{6} + 431 p^{2} T^{7} + 195 p^{6} T^{8} + 161 p^{6} T^{9} + 47 p^{11} T^{10} + 7 p^{12} T^{11} + 15 p^{15} T^{12} + p^{21} T^{14} \)
5 \( 1 + 2 T + 531 T^{2} + 1528 T^{3} + 140618 T^{4} + 470368 T^{5} + 4866676 p T^{6} + 78047332 T^{7} + 4866676 p^{4} T^{8} + 470368 p^{6} T^{9} + 140618 p^{9} T^{10} + 1528 p^{12} T^{11} + 531 p^{15} T^{12} + 2 p^{18} T^{13} + p^{21} T^{14} \)
7 \( 1 + 59 T + 2556 T^{2} + 87863 T^{3} + 2541769 T^{4} + 63434076 T^{5} + 1405832938 T^{6} + 27577593188 T^{7} + 1405832938 p^{3} T^{8} + 63434076 p^{6} T^{9} + 2541769 p^{9} T^{10} + 87863 p^{12} T^{11} + 2556 p^{15} T^{12} + 59 p^{18} T^{13} + p^{21} T^{14} \)
11 \( 1 + 5 T + 3755 T^{2} - 58012 T^{3} + 7669880 T^{4} - 166104440 T^{5} + 14988113598 T^{6} - 215036180534 T^{7} + 14988113598 p^{3} T^{8} - 166104440 p^{6} T^{9} + 7669880 p^{9} T^{10} - 58012 p^{12} T^{11} + 3755 p^{15} T^{12} + 5 p^{18} T^{13} + p^{21} T^{14} \)
13 \( 1 + 67 T + 10237 T^{2} + 587458 T^{3} + 51081852 T^{4} + 2457815768 T^{5} + 162767713622 T^{6} + 6518070060700 T^{7} + 162767713622 p^{3} T^{8} + 2457815768 p^{6} T^{9} + 51081852 p^{9} T^{10} + 587458 p^{12} T^{11} + 10237 p^{15} T^{12} + 67 p^{18} T^{13} + p^{21} T^{14} \)
17 \( 1 + 23 T + 11208 T^{2} + 144569 T^{3} + 78870795 T^{4} + 1707054984 T^{5} + 495290113612 T^{6} + 12833160396024 T^{7} + 495290113612 p^{3} T^{8} + 1707054984 p^{6} T^{9} + 78870795 p^{9} T^{10} + 144569 p^{12} T^{11} + 11208 p^{15} T^{12} + 23 p^{18} T^{13} + p^{21} T^{14} \)
19 \( 1 + 176 T + 40840 T^{2} + 258698 p T^{3} + 707803049 T^{4} + 66775678036 T^{5} + 7310272649602 T^{6} + 562387864513372 T^{7} + 7310272649602 p^{3} T^{8} + 66775678036 p^{6} T^{9} + 707803049 p^{9} T^{10} + 258698 p^{13} T^{11} + 40840 p^{15} T^{12} + 176 p^{18} T^{13} + p^{21} T^{14} \)
23 \( 1 + 218 T + 66974 T^{2} + 10590582 T^{3} + 1949033733 T^{4} + 248040640704 T^{5} + 34723050915604 T^{6} + 3688690538156976 T^{7} + 34723050915604 p^{3} T^{8} + 248040640704 p^{6} T^{9} + 1949033733 p^{9} T^{10} + 10590582 p^{12} T^{11} + 66974 p^{15} T^{12} + 218 p^{18} T^{13} + p^{21} T^{14} \)
29 \( 1 - 168 T + 117046 T^{2} - 22973232 T^{3} + 6768820131 T^{4} - 1293071735524 T^{5} + 249411607198168 T^{6} - 40624497332478372 T^{7} + 249411607198168 p^{3} T^{8} - 1293071735524 p^{6} T^{9} + 6768820131 p^{9} T^{10} - 22973232 p^{12} T^{11} + 117046 p^{15} T^{12} - 168 p^{18} T^{13} + p^{21} T^{14} \)
31 \( 1 + 604 T + 274 p^{2} T^{2} + 77976744 T^{3} + 19722937359 T^{4} + 4058440027390 T^{5} + 782737705298068 T^{6} + 135501449546298300 T^{7} + 782737705298068 p^{3} T^{8} + 4058440027390 p^{6} T^{9} + 19722937359 p^{9} T^{10} + 77976744 p^{12} T^{11} + 274 p^{17} T^{12} + 604 p^{18} T^{13} + p^{21} T^{14} \)
37 \( 1 + 505 T + 302558 T^{2} + 97308201 T^{3} + 36343170681 T^{4} + 9078549401160 T^{5} + 2630319025050828 T^{6} + 545568365760499338 T^{7} + 2630319025050828 p^{3} T^{8} + 9078549401160 p^{6} T^{9} + 36343170681 p^{9} T^{10} + 97308201 p^{12} T^{11} + 302558 p^{15} T^{12} + 505 p^{18} T^{13} + p^{21} T^{14} \)
41 \( 1 + 265 T + 146332 T^{2} + 25771907 T^{3} + 12705640107 T^{4} + 2149799362632 T^{5} + 994998362772976 T^{6} + 174631643209739920 T^{7} + 994998362772976 p^{3} T^{8} + 2149799362632 p^{6} T^{9} + 12705640107 p^{9} T^{10} + 25771907 p^{12} T^{11} + 146332 p^{15} T^{12} + 265 p^{18} T^{13} + p^{21} T^{14} \)
43 \( 1 + 493 T + 385335 T^{2} + 148418932 T^{3} + 71462993408 T^{4} + 23077724590076 T^{5} + 8417547285918154 T^{6} + 2254904231915314798 T^{7} + 8417547285918154 p^{3} T^{8} + 23077724590076 p^{6} T^{9} + 71462993408 p^{9} T^{10} + 148418932 p^{12} T^{11} + 385335 p^{15} T^{12} + 493 p^{18} T^{13} + p^{21} T^{14} \)
47 \( 1 + 244 T + 205374 T^{2} + 14124408 T^{3} + 39309220961 T^{4} + 5784873241192 T^{5} + 5064761044759688 T^{6} + 282690325611779480 T^{7} + 5064761044759688 p^{3} T^{8} + 5784873241192 p^{6} T^{9} + 39309220961 p^{9} T^{10} + 14124408 p^{12} T^{11} + 205374 p^{15} T^{12} + 244 p^{18} T^{13} + p^{21} T^{14} \)
53 \( 1 - 686 T + 782335 T^{2} - 430858628 T^{3} + 295825190954 T^{4} - 133146979625036 T^{5} + 67909011112856856 T^{6} - 24810537240265763492 T^{7} + 67909011112856856 p^{3} T^{8} - 133146979625036 p^{6} T^{9} + 295825190954 p^{9} T^{10} - 430858628 p^{12} T^{11} + 782335 p^{15} T^{12} - 686 p^{18} T^{13} + p^{21} T^{14} \)
61 \( 1 + 838 T + 288966 T^{2} + 50026472 T^{3} + 133404870989 T^{4} + 98124191348950 T^{5} + 30042382947453074 T^{6} + 2970820759019616580 T^{7} + 30042382947453074 p^{3} T^{8} + 98124191348950 p^{6} T^{9} + 133404870989 p^{9} T^{10} + 50026472 p^{12} T^{11} + 288966 p^{15} T^{12} + 838 p^{18} T^{13} + p^{21} T^{14} \)
67 \( 1 + 1504 T + 2353711 T^{2} + 2144225144 T^{3} + 1964485259064 T^{4} + 1312091682415872 T^{5} + 900300407756189746 T^{6} + \)\(48\!\cdots\!24\)\( T^{7} + 900300407756189746 p^{3} T^{8} + 1312091682415872 p^{6} T^{9} + 1964485259064 p^{9} T^{10} + 2144225144 p^{12} T^{11} + 2353711 p^{15} T^{12} + 1504 p^{18} T^{13} + p^{21} T^{14} \)
71 \( 1 + 1267 T + 2739935 T^{2} + 2533363858 T^{3} + 3068481782040 T^{4} + 2177693230263836 T^{5} + 1862252832004989974 T^{6} + \)\(10\!\cdots\!68\)\( T^{7} + 1862252832004989974 p^{3} T^{8} + 2177693230263836 p^{6} T^{9} + 3068481782040 p^{9} T^{10} + 2533363858 p^{12} T^{11} + 2739935 p^{15} T^{12} + 1267 p^{18} T^{13} + p^{21} T^{14} \)
73 \( 1 + 666 T + 1519652 T^{2} + 1042909528 T^{3} + 1182093837125 T^{4} + 800529964727498 T^{5} + 621575126384997598 T^{6} + \)\(38\!\cdots\!40\)\( T^{7} + 621575126384997598 p^{3} T^{8} + 800529964727498 p^{6} T^{9} + 1182093837125 p^{9} T^{10} + 1042909528 p^{12} T^{11} + 1519652 p^{15} T^{12} + 666 p^{18} T^{13} + p^{21} T^{14} \)
79 \( 1 + 2741 T + 5467423 T^{2} + 7269896664 T^{3} + 8152210491056 T^{4} + 7328809929081032 T^{5} + 6034063916206763246 T^{6} + \)\(43\!\cdots\!66\)\( T^{7} + 6034063916206763246 p^{3} T^{8} + 7328809929081032 p^{6} T^{9} + 8152210491056 p^{9} T^{10} + 7269896664 p^{12} T^{11} + 5467423 p^{15} T^{12} + 2741 p^{18} T^{13} + p^{21} T^{14} \)
83 \( 1 + 2025 T + 4269546 T^{2} + 5601527587 T^{3} + 7015070987613 T^{4} + 6855910821135640 T^{5} + 6365067670679753624 T^{6} + \)\(49\!\cdots\!00\)\( T^{7} + 6365067670679753624 p^{3} T^{8} + 6855910821135640 p^{6} T^{9} + 7015070987613 p^{9} T^{10} + 5601527587 p^{12} T^{11} + 4269546 p^{15} T^{12} + 2025 p^{18} T^{13} + p^{21} T^{14} \)
89 \( 1 - 616 T + 4055022 T^{2} - 2213045322 T^{3} + 7491654502867 T^{4} - 3555783851609332 T^{5} + 8226880123952366402 T^{6} - \)\(32\!\cdots\!20\)\( T^{7} + 8226880123952366402 p^{3} T^{8} - 3555783851609332 p^{6} T^{9} + 7491654502867 p^{9} T^{10} - 2213045322 p^{12} T^{11} + 4055022 p^{15} T^{12} - 616 p^{18} T^{13} + p^{21} T^{14} \)
97 \( 1 + 1298 T + 4572619 T^{2} + 4306774598 T^{3} + 9280313142992 T^{4} + 7265084132300450 T^{5} + 12215218568411099932 T^{6} + \)\(81\!\cdots\!96\)\( T^{7} + 12215218568411099932 p^{3} T^{8} + 7265084132300450 p^{6} T^{9} + 9280313142992 p^{9} T^{10} + 4306774598 p^{12} T^{11} + 4572619 p^{15} T^{12} + 1298 p^{18} T^{13} + p^{21} T^{14} \)
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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

−6.21312739133989569841835040733, −6.14939661296469118237415473262, −6.08959625881058652956953784163, −5.93170950683759556616019645673, −5.91820165093941836294734707383, −5.56035653165172438333234127124, −5.35442008171096690081168049259, −5.21201927568492211947405715683, −5.05287762622093793736479334131, −4.98330391686472106529166112028, −4.86399705481368748169217193795, −4.29183320713496553631172998655, −4.19778645602240867787858310538, −4.13406633749339021797690890030, −3.91013703573279453567969678975, −3.88879478007538630779664720436, −3.85620392315917456670127358869, −3.36500429690926176582058097010, −3.05841182362752964719482326657, −2.64189293885785430780595029057, −2.60567859680536277581454774684, −1.99700688626211368875194327715, −1.57937202228929353460327924591, −1.57054288843129401275024617155, −1.50395635050927574200838444523, 0, 0, 0, 0, 0, 0, 0, 1.50395635050927574200838444523, 1.57054288843129401275024617155, 1.57937202228929353460327924591, 1.99700688626211368875194327715, 2.60567859680536277581454774684, 2.64189293885785430780595029057, 3.05841182362752964719482326657, 3.36500429690926176582058097010, 3.85620392315917456670127358869, 3.88879478007538630779664720436, 3.91013703573279453567969678975, 4.13406633749339021797690890030, 4.19778645602240867787858310538, 4.29183320713496553631172998655, 4.86399705481368748169217193795, 4.98330391686472106529166112028, 5.05287762622093793736479334131, 5.21201927568492211947405715683, 5.35442008171096690081168049259, 5.56035653165172438333234127124, 5.91820165093941836294734707383, 5.93170950683759556616019645673, 6.08959625881058652956953784163, 6.14939661296469118237415473262, 6.21312739133989569841835040733

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

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