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Properties

Label 12-10e6-1.1-c11e6-0-0
Degree $12$
Conductor $1000000$
Sign $1$
Analytic cond. $205746.$
Root an. cond. $2.77190$
Motivic weight $11$
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  − 3.07e3·4-s + 530·5-s + 2.83e5·9-s − 6.42e5·11-s + 6.29e6·16-s − 2.41e7·19-s − 1.62e6·20-s − 9.07e7·25-s − 2.56e8·29-s + 4.58e8·31-s − 8.70e8·36-s − 1.64e8·41-s + 1.97e9·44-s + 1.50e8·45-s + 5.59e9·49-s − 3.40e8·55-s + 1.76e10·59-s − 5.02e9·61-s − 1.07e10·64-s + 5.67e10·71-s + 7.40e10·76-s + 2.60e9·79-s + 3.33e9·80-s + 1.64e10·81-s + 2.49e11·89-s − 1.27e10·95-s − 1.82e11·99-s + ⋯
L(s)  = 1  − 3/2·4-s + 0.0758·5-s + 1.59·9-s − 1.20·11-s + 3/2·16-s − 2.23·19-s − 0.113·20-s − 1.85·25-s − 2.32·29-s + 2.87·31-s − 2.39·36-s − 0.222·41-s + 1.80·44-s + 0.121·45-s + 2.82·49-s − 0.0912·55-s + 3.21·59-s − 0.761·61-s − 5/4·64-s + 3.73·71-s + 3.35·76-s + 0.0951·79-s + 0.113·80-s + 0.523·81-s + 4.73·89-s − 0.169·95-s − 1.92·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000000 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000000 ^{s/2} \, \Gamma_{\C}(s+11/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(1000000\)    =    \(2^{6} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(205746.\)
Root analytic conductor: \(2.77190\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 1000000,\ (\ :[11/2]^{6}),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(1.562612576\)
\(L(\frac12)\) \(\approx\) \(1.562612576\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{10} T^{2} )^{3} \)
5 \( 1 - 106 p T + 727999 p^{3} T^{2} + 1256292 p^{7} T^{3} + 727999 p^{14} T^{4} - 106 p^{23} T^{5} + p^{33} T^{6} \)
good3 \( 1 - 283430 T^{2} + 788808167 p^{4} T^{4} - 2219743972340 p^{8} T^{6} + 788808167 p^{26} T^{8} - 283430 p^{44} T^{10} + p^{66} T^{12} \)
7 \( 1 - 5594223150 T^{2} + 282174762670457103 p^{2} T^{4} - \)\(10\!\cdots\!00\)\( p^{4} T^{6} + 282174762670457103 p^{24} T^{8} - 5594223150 p^{44} T^{10} + p^{66} T^{12} \)
11 \( ( 1 + 321364 T + 451069377065 T^{2} + 68139027321485880 T^{3} + 451069377065 p^{11} T^{4} + 321364 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
13 \( 1 - 577710785670 p T^{2} + \)\(27\!\cdots\!07\)\( T^{4} - \)\(60\!\cdots\!80\)\( T^{6} + \)\(27\!\cdots\!07\)\( p^{22} T^{8} - 577710785670 p^{45} T^{10} + p^{66} T^{12} \)
17 \( 1 - 132432185976870 T^{2} + \)\(83\!\cdots\!67\)\( T^{4} - \)\(33\!\cdots\!60\)\( T^{6} + \)\(83\!\cdots\!67\)\( p^{22} T^{8} - 132432185976870 p^{44} T^{10} + p^{66} T^{12} \)
19 \( ( 1 + 12054540 T + 30008106286257 T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + 30008106286257 p^{11} T^{4} + 12054540 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
23 \( 1 - 4235045141210190 T^{2} + \)\(36\!\cdots\!69\)\( p T^{4} - \)\(10\!\cdots\!20\)\( T^{6} + \)\(36\!\cdots\!69\)\( p^{23} T^{8} - 4235045141210190 p^{44} T^{10} + p^{66} T^{12} \)
29 \( ( 1 + 128204910 T + 22822131113381787 T^{2} + \)\(33\!\cdots\!80\)\( T^{3} + 22822131113381787 p^{11} T^{4} + 128204910 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
31 \( ( 1 - 229240896 T + 64386954291598365 T^{2} - \)\(90\!\cdots\!20\)\( T^{3} + 64386954291598365 p^{11} T^{4} - 229240896 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
37 \( 1 - 345211813058554110 T^{2} + \)\(91\!\cdots\!07\)\( T^{4} - \)\(18\!\cdots\!80\)\( T^{6} + \)\(91\!\cdots\!07\)\( p^{22} T^{8} - 345211813058554110 p^{44} T^{10} + p^{66} T^{12} \)
41 \( ( 1 + 82384474 T + 1346507617380213815 T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + 1346507617380213815 p^{11} T^{4} + 82384474 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
43 \( 1 - 2824235125166049750 T^{2} + \)\(42\!\cdots\!47\)\( T^{4} - \)\(44\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!47\)\( p^{22} T^{8} - 2824235125166049750 p^{44} T^{10} + p^{66} T^{12} \)
47 \( 1 - 4774521242059704030 T^{2} + \)\(20\!\cdots\!27\)\( T^{4} - \)\(47\!\cdots\!40\)\( T^{6} + \)\(20\!\cdots\!27\)\( p^{22} T^{8} - 4774521242059704030 p^{44} T^{10} + p^{66} T^{12} \)
53 \( 1 - 32371227025452806430 T^{2} + \)\(55\!\cdots\!27\)\( T^{4} - \)\(63\!\cdots\!40\)\( T^{6} + \)\(55\!\cdots\!27\)\( p^{22} T^{8} - 32371227025452806430 p^{44} T^{10} + p^{66} T^{12} \)
59 \( ( 1 - 8831981180 T + 96619062073692665177 T^{2} - \)\(50\!\cdots\!40\)\( T^{3} + 96619062073692665177 p^{11} T^{4} - 8831981180 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
61 \( ( 1 + 2510396214 T + \)\(12\!\cdots\!15\)\( T^{2} + \)\(21\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!15\)\( p^{11} T^{4} + 2510396214 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
67 \( 1 - \)\(21\!\cdots\!70\)\( T^{2} + \)\(31\!\cdots\!67\)\( T^{4} - \)\(49\!\cdots\!60\)\( T^{6} + \)\(31\!\cdots\!67\)\( p^{22} T^{8} - \)\(21\!\cdots\!70\)\( p^{44} T^{10} + p^{66} T^{12} \)
71 \( ( 1 - 28394209416 T + \)\(79\!\cdots\!65\)\( T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!65\)\( p^{11} T^{4} - 28394209416 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
73 \( 1 - \)\(14\!\cdots\!90\)\( T^{2} + \)\(10\!\cdots\!87\)\( T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!87\)\( p^{22} T^{8} - \)\(14\!\cdots\!90\)\( p^{44} T^{10} + p^{66} T^{12} \)
79 \( ( 1 - 1301275440 T - \)\(13\!\cdots\!63\)\( T^{2} + \)\(24\!\cdots\!80\)\( T^{3} - \)\(13\!\cdots\!63\)\( p^{11} T^{4} - 1301275440 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
83 \( 1 - \)\(32\!\cdots\!70\)\( T^{2} + \)\(56\!\cdots\!67\)\( T^{4} - \)\(71\!\cdots\!60\)\( T^{6} + \)\(56\!\cdots\!67\)\( p^{22} T^{8} - \)\(32\!\cdots\!70\)\( p^{44} T^{10} + p^{66} T^{12} \)
89 \( ( 1 - 124724206270 T + \)\(86\!\cdots\!67\)\( T^{2} - \)\(49\!\cdots\!60\)\( T^{3} + \)\(86\!\cdots\!67\)\( p^{11} T^{4} - 124724206270 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
97 \( 1 - \)\(18\!\cdots\!30\)\( T^{2} + \)\(17\!\cdots\!27\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{6} + \)\(17\!\cdots\!27\)\( p^{22} T^{8} - \)\(18\!\cdots\!30\)\( p^{44} T^{10} + p^{66} T^{12} \)
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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

−9.803585997656866850439421241991, −9.396906711323708128250127533502, −9.202683005798838863213340954503, −8.606167586940627935121242023049, −8.400150554406774795262785234454, −8.310552652932346004264563907811, −7.70966936221151784144915269511, −7.59135748171717876928684857289, −7.26241876915768547059773176496, −6.61523985459222438030033949131, −6.46740506085338644090708872880, −5.95825738353647271869643663954, −5.47718555325092305754895097621, −5.31332013620488253374712914398, −4.69684669438466400053792838785, −4.49654193529319467939199623165, −4.04762458137670081151979365843, −3.77027776488121708317421053139, −3.55183640742035152665427837019, −2.38659173566895437058497228650, −2.29550708362711598235268447601, −1.92237251500110457091818055301, −1.05016682048521973650662562349, −0.68642359667407362647511778914, −0.28653568801652128920958238293, 0.28653568801652128920958238293, 0.68642359667407362647511778914, 1.05016682048521973650662562349, 1.92237251500110457091818055301, 2.29550708362711598235268447601, 2.38659173566895437058497228650, 3.55183640742035152665427837019, 3.77027776488121708317421053139, 4.04762458137670081151979365843, 4.49654193529319467939199623165, 4.69684669438466400053792838785, 5.31332013620488253374712914398, 5.47718555325092305754895097621, 5.95825738353647271869643663954, 6.46740506085338644090708872880, 6.61523985459222438030033949131, 7.26241876915768547059773176496, 7.59135748171717876928684857289, 7.70966936221151784144915269511, 8.310552652932346004264563907811, 8.400150554406774795262785234454, 8.606167586940627935121242023049, 9.202683005798838863213340954503, 9.396906711323708128250127533502, 9.803585997656866850439421241991

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

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