Properties

Label 12-5e12-1.1-c9e6-0-0
Degree $12$
Conductor $244140625$
Sign $1$
Analytic cond. $4.55685\times 10^{6}$
Root an. cond. $3.58829$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.19e3·4-s + 815·9-s − 1.09e5·11-s + 2.77e5·16-s − 1.63e6·19-s − 4.35e6·29-s + 8.54e6·31-s + 9.73e5·36-s + 1.18e7·41-s − 1.30e8·44-s + 1.27e8·49-s − 1.13e7·59-s + 2.50e8·61-s − 4.12e8·64-s + 5.95e8·71-s − 1.95e9·76-s + 6.20e8·79-s + 1.26e8·81-s + 2.20e9·89-s − 8.91e7·99-s − 9.16e8·101-s − 4.25e9·109-s − 5.19e9·116-s − 2.65e9·121-s + 1.02e10·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 2.33·4-s + 0.0414·9-s − 2.25·11-s + 1.05·16-s − 2.88·19-s − 1.14·29-s + 1.66·31-s + 0.0966·36-s + 0.655·41-s − 5.25·44-s + 3.15·49-s − 0.121·59-s + 2.31·61-s − 3.07·64-s + 2.77·71-s − 6.72·76-s + 1.79·79-s + 0.326·81-s + 3.72·89-s − 0.0932·99-s − 0.876·101-s − 2.88·109-s − 2.66·116-s − 1.12·121-s + 3.88·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(5^{12}\)
Sign: $1$
Analytic conductor: \(4.55685\times 10^{6}\)
Root analytic conductor: \(3.58829\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 5^{12} ,\ ( \ : [9/2]^{6} ),\ 1 )\)

Particular Values

\(L(5)\) \(\approx\) \(0.2760096172\)
\(L(\frac12)\) \(\approx\) \(0.2760096172\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 - 1195 T^{2} + 287733 p^{2} T^{4} - 9864815 p^{6} T^{6} + 287733 p^{20} T^{8} - 1195 p^{36} T^{10} + p^{54} T^{12} \)
3 \( 1 - 815 T^{2} - 13997962 p^{2} T^{4} - 82628689595 p^{4} T^{6} - 13997962 p^{20} T^{8} - 815 p^{36} T^{10} + p^{54} T^{12} \)
7 \( 1 - 127514750 T^{2} + 154108736216303 p^{2} T^{4} - \)\(13\!\cdots\!00\)\( p^{4} T^{6} + 154108736216303 p^{20} T^{8} - 127514750 p^{36} T^{10} + p^{54} T^{12} \)
11 \( ( 1 + 54699 T + 5816187440 T^{2} + 182671409832855 T^{3} + 5816187440 p^{9} T^{4} + 54699 p^{18} T^{5} + p^{27} T^{6} )^{2} \)
13 \( 1 - 15844411230 T^{2} + \)\(30\!\cdots\!87\)\( T^{4} - \)\(30\!\cdots\!40\)\( T^{6} + \)\(30\!\cdots\!87\)\( p^{18} T^{8} - 15844411230 p^{36} T^{10} + p^{54} T^{12} \)
17 \( 1 - 597847713835 T^{2} + \)\(15\!\cdots\!02\)\( T^{4} - \)\(24\!\cdots\!55\)\( T^{6} + \)\(15\!\cdots\!02\)\( p^{18} T^{8} - 597847713835 p^{36} T^{10} + p^{54} T^{12} \)
19 \( ( 1 + 818845 T + 468173656712 T^{2} + 302339390932836385 T^{3} + 468173656712 p^{9} T^{4} + 818845 p^{18} T^{5} + p^{27} T^{6} )^{2} \)
23 \( 1 - 215760622290 p T^{2} + \)\(14\!\cdots\!07\)\( T^{4} - \)\(32\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!07\)\( p^{18} T^{8} - 215760622290 p^{37} T^{10} + p^{54} T^{12} \)
29 \( ( 1 + 2175480 T + 1119231270883 p T^{2} + 59155485309271560240 T^{3} + 1119231270883 p^{10} T^{4} + 2175480 p^{18} T^{5} + p^{27} T^{6} )^{2} \)
31 \( ( 1 - 4274066 T + 82851493809465 T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + 82851493809465 p^{9} T^{4} - 4274066 p^{18} T^{5} + p^{27} T^{6} )^{2} \)
37 \( 1 - 575490501959730 T^{2} + \)\(15\!\cdots\!87\)\( T^{4} - \)\(24\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!87\)\( p^{18} T^{8} - 575490501959730 p^{36} T^{10} + p^{54} T^{12} \)
41 \( ( 1 - 5926311 T + 232043727124790 T^{2} - \)\(23\!\cdots\!95\)\( T^{3} + 232043727124790 p^{9} T^{4} - 5926311 p^{18} T^{5} + p^{27} T^{6} )^{2} \)
43 \( 1 - 2755277785986450 T^{2} + \)\(32\!\cdots\!47\)\( T^{4} - \)\(11\!\cdots\!00\)\( p^{2} T^{6} + \)\(32\!\cdots\!47\)\( p^{18} T^{8} - 2755277785986450 p^{36} T^{10} + p^{54} T^{12} \)
47 \( 1 - 4564445424701290 T^{2} + \)\(10\!\cdots\!67\)\( T^{4} - \)\(14\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!67\)\( p^{18} T^{8} - 4564445424701290 p^{36} T^{10} + p^{54} T^{12} \)
53 \( 1 - 830296241726290 T^{2} + \)\(19\!\cdots\!67\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!67\)\( p^{18} T^{8} - 830296241726290 p^{36} T^{10} + p^{54} T^{12} \)
59 \( ( 1 + 5670960 T + 19337695182532817 T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + 19337695182532817 p^{9} T^{4} + 5670960 p^{18} T^{5} + p^{27} T^{6} )^{2} \)
61 \( ( 1 - 125306926 T + 22192130203358915 T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + 22192130203358915 p^{9} T^{4} - 125306926 p^{18} T^{5} + p^{27} T^{6} )^{2} \)
67 \( 1 - 53234092789742135 T^{2} + \)\(12\!\cdots\!02\)\( T^{4} - \)\(21\!\cdots\!55\)\( T^{6} + \)\(12\!\cdots\!02\)\( p^{18} T^{8} - 53234092789742135 p^{36} T^{10} + p^{54} T^{12} \)
71 \( ( 1 - 297550596 T + 87036096018332165 T^{2} - \)\(15\!\cdots\!20\)\( T^{3} + 87036096018332165 p^{9} T^{4} - 297550596 p^{18} T^{5} + p^{27} T^{6} )^{2} \)
73 \( 1 - 155852935645686795 T^{2} + \)\(17\!\cdots\!82\)\( T^{4} - \)\(11\!\cdots\!35\)\( T^{6} + \)\(17\!\cdots\!82\)\( p^{18} T^{8} - 155852935645686795 p^{36} T^{10} + p^{54} T^{12} \)
79 \( ( 1 - 310025170 T + 176092553119892457 T^{2} - \)\(48\!\cdots\!60\)\( T^{3} + 176092553119892457 p^{9} T^{4} - 310025170 p^{18} T^{5} + p^{27} T^{6} )^{2} \)
83 \( 1 - 836375861147057535 T^{2} + \)\(32\!\cdots\!02\)\( T^{4} - \)\(77\!\cdots\!55\)\( T^{6} + \)\(32\!\cdots\!02\)\( p^{18} T^{8} - 836375861147057535 p^{36} T^{10} + p^{54} T^{12} \)
89 \( ( 1 - 1103860035 T + 1320664213408319502 T^{2} - \)\(75\!\cdots\!55\)\( T^{3} + 1320664213408319502 p^{9} T^{4} - 1103860035 p^{18} T^{5} + p^{27} T^{6} )^{2} \)
97 \( 1 - 2865108733311184890 T^{2} + \)\(43\!\cdots\!67\)\( T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(43\!\cdots\!67\)\( p^{18} T^{8} - 2865108733311184890 p^{36} T^{10} + p^{54} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.021727425202398088449442354075, −7.60005597542704149604360686815, −7.58665471166831721862882613091, −7.37397584050575960425352729585, −6.73398100316321392349052534716, −6.73314646366840056569185993614, −6.52221893871750887945163231796, −6.30868199971019862978104645118, −6.04068486171003362582004589949, −5.52797801513244824277603784294, −5.30567983365970725671355808252, −5.03948903960938762892291819319, −4.71079106191059034304430468232, −4.17010621746123734304359606362, −3.90691337546193740276913612751, −3.76744448635663106736258238523, −2.92158998976298554817300613944, −2.70919388528199052686863164702, −2.39394081384911155270807133389, −2.21909790519304437450975363184, −2.14874760932246224980062611409, −1.77405655959351973659783273008, −0.885837628348138331128736733116, −0.74552763452892293371668596158, −0.05926775966929461258012590304, 0.05926775966929461258012590304, 0.74552763452892293371668596158, 0.885837628348138331128736733116, 1.77405655959351973659783273008, 2.14874760932246224980062611409, 2.21909790519304437450975363184, 2.39394081384911155270807133389, 2.70919388528199052686863164702, 2.92158998976298554817300613944, 3.76744448635663106736258238523, 3.90691337546193740276913612751, 4.17010621746123734304359606362, 4.71079106191059034304430468232, 5.03948903960938762892291819319, 5.30567983365970725671355808252, 5.52797801513244824277603784294, 6.04068486171003362582004589949, 6.30868199971019862978104645118, 6.52221893871750887945163231796, 6.73314646366840056569185993614, 6.73398100316321392349052534716, 7.37397584050575960425352729585, 7.58665471166831721862882613091, 7.60005597542704149604360686815, 8.021727425202398088449442354075

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

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