↑

Properties

Label 2-31e2-31.4-c1-0-58
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.108 + 0.334i)2-s + (0.894 − 2.75i)3-s + (1.51 − 1.10i)4-s + 2.97·5-s + 1.01·6-s + (0.875 − 0.636i)7-s + (1.10 + 0.800i)8-s + (−4.34 − 3.15i)9-s + (0.322 + 0.993i)10-s + (−1.96 + 1.42i)11-s + (−1.67 − 5.16i)12-s + (−0.888 + 2.73i)13-s + (0.307 + 0.223i)14-s + (2.65 − 8.18i)15-s + (1.01 − 3.11i)16-s + (−1.47 − 1.07i)17-s + ⋯
L(s)  = 1  + (0.0767 + 0.236i)2-s + (0.516 − 1.58i)3-s + (0.759 − 0.551i)4-s + 1.32·5-s + 0.415·6-s + (0.330 − 0.240i)7-s + (0.389 + 0.283i)8-s + (−1.44 − 1.05i)9-s + (0.102 + 0.314i)10-s + (−0.593 + 0.430i)11-s + (−0.484 − 1.49i)12-s + (−0.246 + 0.757i)13-s + (0.0822 + 0.0597i)14-s + (0.686 − 2.11i)15-s + (0.252 − 0.778i)16-s + (−0.358 − 0.260i)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

\(L(1)\) \(\approx\) \(2.09090 - 1.98381i\)
\(L(\frac12)\) \(\approx\) \(2.09090 - 1.98381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.108 - 0.334i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.894 + 2.75i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - 2.97T + 5T^{2} \)
7 \( 1 + (-0.875 + 0.636i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (1.96 - 1.42i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.888 - 2.73i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.47 + 1.07i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.654 - 2.01i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.357 + 0.259i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.976 - 3.00i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 3.14T + 37T^{2} \)
41 \( 1 + (2.14 + 6.60i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-2.59 - 8.00i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (2.46 - 7.57i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.02 + 2.92i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.71 - 11.4i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 + 6.43T + 67T^{2} \)
71 \( 1 + (1.35 + 0.986i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-11.6 + 8.44i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.78 + 2.02i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.97 + 12.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-1.82 + 1.32i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.76 - 2.00i)T + (29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.788528022391719532682922357013, −8.923309318994388018727684652505, −7.81720798681495547633333423788, −7.21371684579672912369741841978, −6.49102185483813093984910504024, −5.86609839464488134842289688759, −4.86118482612306172529209119203, −2.78748675614598189006727887749, −2.00500301367584241673982260448, −1.38627562616953782029989164842, 2.11976821574056237037964136279, 2.85392134828851024265920624489, 3.77522617796535347956926283481, 5.01082399486813927117271898153, 5.61622343285367379816707726238, 6.73161865030371105480980530527, 8.081903585794718121243328795347, 8.621265522573228413691956200452, 9.685744452522065943413719842000, 10.14614349009165218059312622003

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