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Properties

Label 2-31e2-31.25-c1-0-6
Degree $2$
Conductor $961$
Sign $-0.198 - 0.980i$
Analytic cond. $7.67362$
Root an. cond. $2.77013$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.85·2-s + (0.249 + 0.432i)3-s + 1.42·4-s + (−0.603 + 1.04i)5-s + (−0.461 − 0.800i)6-s + (−1.86 − 3.23i)7-s + 1.06·8-s + (1.37 − 2.38i)9-s + (1.11 − 1.93i)10-s + (−0.928 + 1.60i)11-s + (0.355 + 0.615i)12-s + (−2.59 + 4.48i)13-s + (3.45 + 5.98i)14-s − 0.602·15-s − 4.82·16-s + (−2.83 − 4.90i)17-s + ⋯
L(s)  = 1  − 1.30·2-s + (0.144 + 0.249i)3-s + 0.711·4-s + (−0.269 + 0.467i)5-s + (−0.188 − 0.326i)6-s + (−0.705 − 1.22i)7-s + 0.377·8-s + (0.458 − 0.794i)9-s + (0.353 − 0.611i)10-s + (−0.279 + 0.484i)11-s + (0.102 + 0.177i)12-s + (−0.718 + 1.24i)13-s + (0.922 + 1.59i)14-s − 0.155·15-s − 1.20·16-s + (−0.687 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252714 + 0.309014i\)
\(L(\frac12)\) \(\approx\) \(0.252714 + 0.309014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + 1.85T + 2T^{2} \)
3 \( 1 + (-0.249 - 0.432i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.603 - 1.04i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.86 + 3.23i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.928 - 1.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.59 - 4.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.83 + 4.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.719 - 1.24i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.54T + 23T^{2} \)
29 \( 1 + 1.37T + 29T^{2} \)
37 \( 1 + (-2.25 - 3.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.35 + 4.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.27 - 5.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.22T + 47T^{2} \)
53 \( 1 + (6.48 - 11.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.09 - 1.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 2.68T + 61T^{2} \)
67 \( 1 + (1.44 - 2.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.50 - 7.79i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.10 - 3.64i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.61 - 9.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.23 - 9.07i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.70T + 89T^{2} \)
97 \( 1 + 8.29T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.862828926338043705462682178213, −9.589791585288934457948494586079, −8.922707727199202482604874014207, −7.57165491959393686960821546309, −7.07336929123276663669307203155, −6.67955401936973859313069516214, −4.71648712025559512005971947763, −4.01939233047940614284784667280, −2.73788507274966792712294464486, −1.11800141854986783003322332205, 0.33022784070307529084258305051, 1.95142847663836772393266232646, 2.95164449655473386545464731845, 4.60413289244142368525219724154, 5.52541137907514497958704250196, 6.62764851913054558555305830892, 7.68362312054657245790486729558, 8.258983709762988350334990367005, 8.856394521657120985661737132243, 9.628795475223198214510514314981

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