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Properties

Label 2-31e2-31.4-c1-0-58
Degree 22
Conductor 961961
Sign 0.0525+0.998i0.0525 + 0.998i
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  + (0.108 + 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

Λ(s)=(961s/2 ΓC(s) L(s)=((0.0525+0.998i) Λ‾(2−s)\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}
Λ(s)=(961s/2 ΓC(s+1/2) L(s)=((0.0525+0.998i) Λ‾(1−s)\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: 22
Conductor: 961961    =    31231^{2}
Sign: 0.0525+0.998i0.0525 + 0.998i
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), 0.0525+0.998i)(2,\ 961,\ (\ :1/2),\ 0.0525 + 0.998i)

Particular Values

L(1)L(1) ≈\approx 2.09090−1.98381i2.09090 - 1.98381i
L(12)L(\frac12) ≈\approx 2.09090−1.98381i2.09090 - 1.98381i
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+(−0.108−0.334i)T+(−1.61+1.17i)T2 1 + (-0.108 - 0.334i)T + (-1.61 + 1.17i)T^{2}
3 1+(−0.894+2.75i)T+(−2.42−1.76i)T2 1 + (-0.894 + 2.75i)T + (-2.42 - 1.76i)T^{2}
5 1−2.97T+5T2 1 - 2.97T + 5T^{2}
7 1+(−0.875+0.636i)T+(2.16−6.65i)T2 1 + (-0.875 + 0.636i)T + (2.16 - 6.65i)T^{2}
11 1+(1.96−1.42i)T+(3.39−10.4i)T2 1 + (1.96 - 1.42i)T + (3.39 - 10.4i)T^{2}
13 1+(0.888−2.73i)T+(−10.5−7.64i)T2 1 + (0.888 - 2.73i)T + (-10.5 - 7.64i)T^{2}
17 1+(1.47+1.07i)T+(5.25+16.1i)T2 1 + (1.47 + 1.07i)T + (5.25 + 16.1i)T^{2}
19 1+(−0.654−2.01i)T+(−15.3+11.1i)T2 1 + (-0.654 - 2.01i)T + (-15.3 + 11.1i)T^{2}
23 1+(0.357+0.259i)T+(7.10+21.8i)T2 1 + (0.357 + 0.259i)T + (7.10 + 21.8i)T^{2}
29 1+(−0.976−3.00i)T+(−23.4+17.0i)T2 1 + (-0.976 - 3.00i)T + (-23.4 + 17.0i)T^{2}
37 1+3.14T+37T2 1 + 3.14T + 37T^{2}
41 1+(2.14+6.60i)T+(−33.1+24.0i)T2 1 + (2.14 + 6.60i)T + (-33.1 + 24.0i)T^{2}
43 1+(−2.59−8.00i)T+(−34.7+25.2i)T2 1 + (-2.59 - 8.00i)T + (-34.7 + 25.2i)T^{2}
47 1+(2.46−7.57i)T+(−38.0−27.6i)T2 1 + (2.46 - 7.57i)T + (-38.0 - 27.6i)T^{2}
53 1+(4.02+2.92i)T+(16.3+50.4i)T2 1 + (4.02 + 2.92i)T + (16.3 + 50.4i)T^{2}
59 1+(3.71−11.4i)T+(−47.7−34.6i)T2 1 + (3.71 - 11.4i)T + (-47.7 - 34.6i)T^{2}
61 1−14.4T+61T2 1 - 14.4T + 61T^{2}
67 1+6.43T+67T2 1 + 6.43T + 67T^{2}
71 1+(1.35+0.986i)T+(21.9+67.5i)T2 1 + (1.35 + 0.986i)T + (21.9 + 67.5i)T^{2}
73 1+(−11.6+8.44i)T+(22.5−69.4i)T2 1 + (-11.6 + 8.44i)T + (22.5 - 69.4i)T^{2}
79 1+(2.78+2.02i)T+(24.4+75.1i)T2 1 + (2.78 + 2.02i)T + (24.4 + 75.1i)T^{2}
83 1+(3.97+12.2i)T+(−67.1+48.7i)T2 1 + (3.97 + 12.2i)T + (-67.1 + 48.7i)T^{2}
89 1+(−1.82+1.32i)T+(27.5−84.6i)T2 1 + (-1.82 + 1.32i)T + (27.5 - 84.6i)T^{2}
97 1+(2.76−2.00i)T+(29.9−92.2i)T2 1 + (2.76 - 2.00i)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

−9.788528022391719532682922357013, −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 ZZ-function along the critical line