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Properties

Label 2-31e2-31.4-c1-0-35
Degree 22
Conductor 961961
Sign −0.634+0.773i-0.634 + 0.773i
Analytic cond. 7.673627.67362
Root an. cond. 2.770132.77013
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.874 + 2.68i)3-s + (0.809 − 0.587i)4-s + 2.82·6-s + (−3.23 + 2.35i)7-s + (−2.42 − 1.76i)8-s + (−4.04 − 2.93i)9-s + (2.28 − 1.66i)11-s + (0.874 + 2.68i)12-s + (−0.437 + 1.34i)13-s + (3.23 + 2.35i)14-s + (−0.309 + 0.951i)16-s + (1.14 + 0.831i)17-s + (−1.54 + 4.75i)18-s + (−1.23 − 3.80i)19-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.504 + 1.55i)3-s + (0.404 − 0.293i)4-s + 1.15·6-s + (−1.22 + 0.888i)7-s + (−0.858 − 0.623i)8-s + (−1.34 − 0.979i)9-s + (0.689 − 0.501i)11-s + (0.252 + 0.776i)12-s + (−0.121 + 0.373i)13-s + (0.864 + 0.628i)14-s + (−0.0772 + 0.237i)16-s + (0.277 + 0.201i)17-s + (−0.364 + 1.12i)18-s + (−0.283 − 0.872i)19-s + ⋯

Functional equation

Λ(s)=(961s/2 ΓC(s) L(s)=((−0.634+0.773i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(961s/2 ΓC(s+1/2) L(s)=((−0.634+0.773i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 961961    =    31231^{2}
Sign: −0.634+0.773i-0.634 + 0.773i
Analytic conductor: 7.673627.67362
Root analytic conductor: 2.770132.77013
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ961(531,⋅)\chi_{961} (531, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 961, ( :1/2), −0.634+0.773i)(2,\ 961,\ (\ :1/2),\ -0.634 + 0.773i)

Particular Values

L(1)L(1) ≈\approx 0.127006−0.268505i0.127006 - 0.268505i
L(12)L(\frac12) ≈\approx 0.127006−0.268505i0.127006 - 0.268505i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad31 1 1
good2 1+(0.309+0.951i)T+(−1.61+1.17i)T2 1 + (0.309 + 0.951i)T + (-1.61 + 1.17i)T^{2}
3 1+(0.874−2.68i)T+(−2.42−1.76i)T2 1 + (0.874 - 2.68i)T + (-2.42 - 1.76i)T^{2}
5 1+5T2 1 + 5T^{2}
7 1+(3.23−2.35i)T+(2.16−6.65i)T2 1 + (3.23 - 2.35i)T + (2.16 - 6.65i)T^{2}
11 1+(−2.28+1.66i)T+(3.39−10.4i)T2 1 + (-2.28 + 1.66i)T + (3.39 - 10.4i)T^{2}
13 1+(0.437−1.34i)T+(−10.5−7.64i)T2 1 + (0.437 - 1.34i)T + (-10.5 - 7.64i)T^{2}
17 1+(−1.14−0.831i)T+(5.25+16.1i)T2 1 + (-1.14 - 0.831i)T + (5.25 + 16.1i)T^{2}
19 1+(1.23+3.80i)T+(−15.3+11.1i)T2 1 + (1.23 + 3.80i)T + (-15.3 + 11.1i)T^{2}
23 1+(4.57+3.32i)T+(7.10+21.8i)T2 1 + (4.57 + 3.32i)T + (7.10 + 21.8i)T^{2}
29 1+(−0.437−1.34i)T+(−23.4+17.0i)T2 1 + (-0.437 - 1.34i)T + (-23.4 + 17.0i)T^{2}
37 1−4.24T+37T2 1 - 4.24T + 37T^{2}
41 1+(0.618+1.90i)T+(−33.1+24.0i)T2 1 + (0.618 + 1.90i)T + (-33.1 + 24.0i)T^{2}
43 1+(2.62+8.06i)T+(−34.7+25.2i)T2 1 + (2.62 + 8.06i)T + (-34.7 + 25.2i)T^{2}
47 1+(−3.70+11.4i)T+(−38.0−27.6i)T2 1 + (-3.70 + 11.4i)T + (-38.0 - 27.6i)T^{2}
53 1+(−3.43−2.49i)T+(16.3+50.4i)T2 1 + (-3.43 - 2.49i)T + (16.3 + 50.4i)T^{2}
59 1+(−2.47+7.60i)T+(−47.7−34.6i)T2 1 + (-2.47 + 7.60i)T + (-47.7 - 34.6i)T^{2}
61 1+1.41T+61T2 1 + 1.41T + 61T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1+(6.47+4.70i)T+(21.9+67.5i)T2 1 + (6.47 + 4.70i)T + (21.9 + 67.5i)T^{2}
73 1+(3.43−2.49i)T+(22.5−69.4i)T2 1 + (3.43 - 2.49i)T + (22.5 - 69.4i)T^{2}
79 1+(9.15+6.65i)T+(24.4+75.1i)T2 1 + (9.15 + 6.65i)T + (24.4 + 75.1i)T^{2}
83 1+(−4.37−13.4i)T+(−67.1+48.7i)T2 1 + (-4.37 - 13.4i)T + (-67.1 + 48.7i)T^{2}
89 1+(5.72−4.15i)T+(27.5−84.6i)T2 1 + (5.72 - 4.15i)T + (27.5 - 84.6i)T^{2}
97 1+(6.47−4.70i)T+(29.9−92.2i)T2 1 + (6.47 - 4.70i)T + (29.9 - 92.2i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−9.912160220449787086122633418382, −9.217543311789550128446599770618, −8.695868073426479530663449270290, −6.81262061253140831896256908294, −6.07255889609660993157711140221, −5.47987596220005335123217619709, −4.10264067400641663459788834061, −3.38012084922844866881492975228, −2.31867655238823765203183921213, −0.15113960954661166720935292978, 1.46741319083444709556948620869, 2.80101347493079498441657035596, 4.00613070358330865625264534731, 5.94297925892233716733500501464, 6.12034666130550408409236376899, 7.04275041098555200846254583736, 7.55024240567844436568302233407, 8.127330590097757818572508212239, 9.455200924240857086833541392578, 10.22918260959923321436381468597

Graph of the ZZ-function along the critical line