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Properties

Label 2-31e2-31.19-c1-0-26
Degree 22
Conductor 961961
Sign −0.999−0.00444i-0.999 - 0.00444i
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  + (−1.30 − 0.951i)2-s + (−2.95 − 1.31i)3-s + (0.190 + 0.587i)4-s + (−0.5 + 0.866i)5-s + (2.61 + 4.53i)6-s + (0.157 − 0.175i)7-s + (−0.690 + 2.12i)8-s + (4.99 + 5.55i)9-s + (1.47 − 0.658i)10-s + (−1.95 − 0.415i)11-s + (0.209 − 1.98i)12-s + (0.338 + 3.21i)13-s + (−0.373 + 0.0794i)14-s + (2.61 − 1.90i)15-s + (3.92 − 2.85i)16-s + (−0.747 + 0.158i)17-s + ⋯
L(s)  = 1  + (−0.925 − 0.672i)2-s + (−1.70 − 0.759i)3-s + (0.0954 + 0.293i)4-s + (−0.223 + 0.387i)5-s + (1.06 + 1.85i)6-s + (0.0597 − 0.0663i)7-s + (−0.244 + 0.751i)8-s + (1.66 + 1.85i)9-s + (0.467 − 0.208i)10-s + (−0.589 − 0.125i)11-s + (0.0603 − 0.574i)12-s + (0.0938 + 0.892i)13-s + (−0.0998 + 0.0212i)14-s + (0.675 − 0.491i)15-s + (0.981 − 0.713i)16-s + (−0.181 + 0.0385i)17-s + ⋯

Functional equation

Λ(s)=(961s/2 ΓC(s) L(s)=((−0.999−0.00444i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00444i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(961s/2 ΓC(s+1/2) L(s)=((−0.999−0.00444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00444i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 961961    =    31231^{2}
Sign: −0.999−0.00444i-0.999 - 0.00444i
Analytic conductor: 7.673627.67362
Root analytic conductor: 2.770132.77013
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ961(732,⋅)\chi_{961} (732, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 961, ( :1/2), −0.999−0.00444i)(2,\ 961,\ (\ :1/2),\ -0.999 - 0.00444i)

Particular Values

L(1)L(1) ≈\approx 0.000462493+0.208206i0.000462493 + 0.208206i
L(12)L(\frac12) ≈\approx 0.000462493+0.208206i0.000462493 + 0.208206i
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+(1.30+0.951i)T+(0.618+1.90i)T2 1 + (1.30 + 0.951i)T + (0.618 + 1.90i)T^{2}
3 1+(2.95+1.31i)T+(2.00+2.22i)T2 1 + (2.95 + 1.31i)T + (2.00 + 2.22i)T^{2}
5 1+(0.5−0.866i)T+(−2.5−4.33i)T2 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2}
7 1+(−0.157+0.175i)T+(−0.731−6.96i)T2 1 + (-0.157 + 0.175i)T + (-0.731 - 6.96i)T^{2}
11 1+(1.95+0.415i)T+(10.0+4.47i)T2 1 + (1.95 + 0.415i)T + (10.0 + 4.47i)T^{2}
13 1+(−0.338−3.21i)T+(−12.7+2.70i)T2 1 + (-0.338 - 3.21i)T + (-12.7 + 2.70i)T^{2}
17 1+(0.747−0.158i)T+(15.5−6.91i)T2 1 + (0.747 - 0.158i)T + (15.5 - 6.91i)T^{2}
19 1+(−0.233+2.22i)T+(−18.5−3.95i)T2 1 + (-0.233 + 2.22i)T + (-18.5 - 3.95i)T^{2}
23 1+(−1.76+5.42i)T+(−18.6−13.5i)T2 1 + (-1.76 + 5.42i)T + (-18.6 - 13.5i)T^{2}
29 1+(2.23+1.62i)T+(8.96+27.5i)T2 1 + (2.23 + 1.62i)T + (8.96 + 27.5i)T^{2}
37 1+(−1−1.73i)T+(−18.5+32.0i)T2 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2}
41 1+(−6.39+2.84i)T+(27.4−30.4i)T2 1 + (-6.39 + 2.84i)T + (27.4 - 30.4i)T^{2}
43 1+(0.129−1.22i)T+(−42.0−8.94i)T2 1 + (0.129 - 1.22i)T + (-42.0 - 8.94i)T^{2}
47 1+(2−1.45i)T+(14.5−44.6i)T2 1 + (2 - 1.45i)T + (14.5 - 44.6i)T^{2}
53 1+(7.00+7.78i)T+(−5.54+52.7i)T2 1 + (7.00 + 7.78i)T + (-5.54 + 52.7i)T^{2}
59 1+(−2.04−0.909i)T+(39.4+43.8i)T2 1 + (-2.04 - 0.909i)T + (39.4 + 43.8i)T^{2}
61 1−8.18T+61T2 1 - 8.18T + 61T^{2}
67 1+(4−6.92i)T+(−33.5−58.0i)T2 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2}
71 1+(6.14+6.82i)T+(−7.42+70.6i)T2 1 + (6.14 + 6.82i)T + (-7.42 + 70.6i)T^{2}
73 1+(8.28+1.76i)T+(66.6+29.6i)T2 1 + (8.28 + 1.76i)T + (66.6 + 29.6i)T^{2}
79 1+(−11.4+2.43i)T+(72.1−32.1i)T2 1 + (-11.4 + 2.43i)T + (72.1 - 32.1i)T^{2}
83 1+(13.6−6.07i)T+(55.5−61.6i)T2 1 + (13.6 - 6.07i)T + (55.5 - 61.6i)T^{2}
89 1+(−3.61−11.1i)T+(−72.0+52.3i)T2 1 + (-3.61 - 11.1i)T + (-72.0 + 52.3i)T^{2}
97 1+(4.92+15.1i)T+(−78.4+57.0i)T2 1 + (4.92 + 15.1i)T + (-78.4 + 57.0i)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

−9.913737553296989845008047448902, −8.912476505579886803445758920083, −7.84733721006105728717279447341, −7.05574185086875396355502102351, −6.27260602718360331329765466662, −5.36238127818924991538182404450, −4.49684487333086695161552711886, −2.58292996710064620528850832071, −1.43357197271161183365410659094, −0.23567692889004931292023260099, 0.938791577748437918209742390881, 3.49471874149431193227568583481, 4.55066861970383148489962242127, 5.46652886372924600501066242669, 6.08700178299702460366128814983, 7.10492888017794513395171620279, 7.85648123916705843052285899481, 8.860907903832274713085315712290, 9.720557659277721424725403276313, 10.30108373006669157371349266687

Graph of the ZZ-function along the critical line