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Properties

Label 1-31e2-961.82-r0-0-0
Degree $1$
Conductor $961$
Sign $-0.868 - 0.495i$
Analytic cond. $4.46286$
Root an. cond. $4.46286$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.511 + 0.859i)2-s + (0.00337 − 0.999i)3-s + (−0.476 − 0.879i)4-s + (0.0843 + 0.996i)5-s + (0.857 + 0.514i)6-s + (0.606 − 0.794i)7-s + (0.999 + 0.0405i)8-s + (−0.999 − 0.00675i)9-s + (−0.899 − 0.437i)10-s + (−0.887 + 0.461i)11-s + (−0.880 + 0.473i)12-s + (0.976 − 0.214i)13-s + (0.372 + 0.928i)14-s + (0.996 − 0.0809i)15-s + (−0.546 + 0.837i)16-s + (−0.633 − 0.773i)17-s + ⋯
L(s)  = 1  + (−0.511 + 0.859i)2-s + (0.00337 − 0.999i)3-s + (−0.476 − 0.879i)4-s + (0.0843 + 0.996i)5-s + (0.857 + 0.514i)6-s + (0.606 − 0.794i)7-s + (0.999 + 0.0405i)8-s + (−0.999 − 0.00675i)9-s + (−0.899 − 0.437i)10-s + (−0.887 + 0.461i)11-s + (−0.880 + 0.473i)12-s + (0.976 − 0.214i)13-s + (0.372 + 0.928i)14-s + (0.996 − 0.0809i)15-s + (−0.546 + 0.837i)16-s + (−0.633 − 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.868 - 0.495i$
Analytic conductor: \(4.46286\)
Root analytic conductor: \(4.46286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 961,\ (0:\ ),\ -0.868 - 0.495i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04994301487 - 0.1884832878i\)
\(L(\frac12)\) \(\approx\) \(0.04994301487 - 0.1884832878i\)
\(L(1)\) \(\approx\) \(0.6274166027 + 0.02345273037i\)
\(L(1)\) \(\approx\) \(0.6274166027 + 0.02345273037i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (-0.511 + 0.859i)T \)
3 \( 1 + (0.00337 - 0.999i)T \)
5 \( 1 + (0.0843 + 0.996i)T \)
7 \( 1 + (0.606 - 0.794i)T \)
11 \( 1 + (-0.887 + 0.461i)T \)
13 \( 1 + (0.976 - 0.214i)T \)
17 \( 1 + (-0.633 - 0.773i)T \)
19 \( 1 + (-0.749 - 0.661i)T \)
23 \( 1 + (-0.289 + 0.957i)T \)
29 \( 1 + (-0.999 + 0.0202i)T \)
37 \( 1 + (-0.315 - 0.948i)T \)
41 \( 1 + (-0.968 - 0.247i)T \)
43 \( 1 + (0.138 + 0.990i)T \)
47 \( 1 + (-0.328 - 0.944i)T \)
53 \( 1 + (0.994 + 0.107i)T \)
59 \( 1 + (-0.302 + 0.953i)T \)
61 \( 1 + (-0.612 - 0.790i)T \)
67 \( 1 + (0.780 + 0.625i)T \)
71 \( 1 + (0.995 - 0.0944i)T \)
73 \( 1 + (-0.961 + 0.273i)T \)
79 \( 1 + (-0.961 - 0.273i)T \)
83 \( 1 + (-0.488 - 0.872i)T \)
89 \( 1 + (-0.853 + 0.520i)T \)
97 \( 1 + (0.902 - 0.431i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.68601610153499214557585457838, −21.13092099162078480703414227228, −20.69771830278681189354785051301, −20.04720889568876643286448450521, −18.90009701413760988959649092938, −18.31565406332270668400676433901, −17.26922401898085772160636651248, −16.7073754830874685807532119497, −15.90562862031449820270928588187, −15.18421091265501437386722885354, −13.99729413184390184063205522867, −13.120350290463191504990420674611, −12.35191607598670154212656430746, −11.441193170880415275320182137, −10.79449813415345046577052216438, −10.04754806761617928526529795325, −8.99297005293491531479856182331, −8.46175567314527499501938608508, −8.133920614495542866998397525151, −6.112994846494001795909348921676, −5.2361236489589018735449600432, −4.406759471853641783640931651742, −3.65890103237048443514548087882, −2.44006394281593112200010162197, −1.56633205548042651537440742252, 0.09739793751075570426239278657, 1.51781079292951530947042695671, 2.40066182455577314527289877255, 3.80481839288412698860078141602, 5.09775871701210251921603314597, 5.96131348100555315586836678058, 6.92842028948537161591646627113, 7.332360638991906828535686732654, 8.054679689339125466987126935809, 8.96582175183919376989921081455, 10.15653704073757818305360983262, 10.94651584203007980456882700400, 11.45197435458861195508332514690, 13.17348842960462666584662106591, 13.485815227272218410379782156489, 14.29204721847235355394265186000, 15.10159315476848470132597467538, 15.81167315160861507491268303510, 16.99359826692700186704330312804, 17.72255299104651664134022612548, 18.15729380253737474466062871540, 18.70010671330691310130325717689, 19.708232678661097988935716475933, 20.30394589280171748447222267465, 21.48838952784709312651111437562

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