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Properties

Label 2-31e2-31.4-c1-0-49
Degree $2$
Conductor $961$
Sign $0.0525 + 0.998i$
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  + (−0.213 − 0.655i)2-s + (0.278 − 0.858i)3-s + (1.23 − 0.896i)4-s + 3.70·5-s − 0.622·6-s + (−0.617 + 0.448i)7-s + (−1.96 − 1.42i)8-s + (1.76 + 1.28i)9-s + (−0.789 − 2.43i)10-s + (−3.32 + 2.41i)11-s + (−0.425 − 1.30i)12-s + (0.899 − 2.76i)13-s + (0.425 + 0.309i)14-s + (1.03 − 3.18i)15-s + (0.424 − 1.30i)16-s + (1.05 + 0.769i)17-s + ⋯
L(s)  = 1  + (−0.150 − 0.463i)2-s + (0.160 − 0.495i)3-s + (0.616 − 0.448i)4-s + 1.65·5-s − 0.254·6-s + (−0.233 + 0.169i)7-s + (−0.695 − 0.505i)8-s + (0.589 + 0.428i)9-s + (−0.249 − 0.768i)10-s + (−1.00 + 0.729i)11-s + (−0.122 − 0.377i)12-s + (0.249 − 0.767i)13-s + (0.113 + 0.0826i)14-s + (0.266 − 0.821i)15-s + (0.106 − 0.326i)16-s + (0.256 + 0.186i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0525 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0525 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.0525 + 0.998i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ 0.0525 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68571 - 1.59937i\)
\(L(\frac12)\) \(\approx\) \(1.68571 - 1.59937i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.213 + 0.655i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.278 + 0.858i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - 3.70T + 5T^{2} \)
7 \( 1 + (0.617 - 0.448i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (3.32 - 2.41i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.899 + 2.76i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.05 - 0.769i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.17 + 3.61i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.65 + 1.92i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.51 + 4.65i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 + (-0.233 - 0.718i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-2.21 - 6.80i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.270 - 0.833i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.89 - 2.10i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.286 + 0.881i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 + 2.08T + 67T^{2} \)
71 \( 1 + (-6.25 - 4.54i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.57 - 3.32i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (11.4 + 8.28i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.35 - 13.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (3.58 - 2.60i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-4.26 + 3.10i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.915757306564232011238613297360, −9.422816833548723504795657072614, −8.095233880530206128673800976436, −7.22711210637073110943246270863, −6.24645620934715100742675047151, −5.75696105920472299166796201272, −4.67558544588175958104240677240, −2.65669696452825218333135857135, −2.33626463801953113023479446943, −1.20402119077108884747539040169, 1.72099140785476511691342866512, 2.80514261730175530961083396966, 3.85026470887513976802235809849, 5.31508189739776233501043310088, 6.02290983632533836622970562626, 6.69429815590218498977403454789, 7.65836006279002550622636081016, 8.689182443546172184743006138482, 9.383091852239930357484920336707, 10.16228516421618274233802459782

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