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Properties

Label 2-31e2-31.18-c1-0-48
Degree 22
Conductor 961961
Sign 0.949+0.314i0.949 + 0.314i
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)=((0.949+0.314i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(961s/2 ΓC(s+1/2) L(s)=((0.949+0.314i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 961961    =    31231^{2}
Sign: 0.949+0.314i0.949 + 0.314i
Analytic conductor: 7.673627.67362
Root analytic conductor: 2.770132.77013
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ961(235,⋅)\chi_{961} (235, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 961, ( :1/2), 0.949+0.314i)(2,\ 961,\ (\ :1/2),\ 0.949 + 0.314i)

Particular Values

L(1)L(1) ≈\approx 1.70496−0.275511i1.70496 - 0.275511i
L(12)L(\frac12) ≈\approx 1.70496−0.275511i1.70496 - 0.275511i
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

−9.484813365600693533121657082539, −8.860709610671396030333403916115, −8.319886547756006381700471053969, −7.80143734952898177262368619920, −6.84453833934405824115446081525, −6.36642021827019285491018423839, −4.35652633187241642517758580828, −3.56023356987468585740590288174, −2.29607635489839945317745321525, −1.02874398411419716296613537054, 1.47557155684217227106171692713, 2.69264545887661440283173261105, 3.31145309853263925104729576062, 4.39582506513835105745450693415, 5.58083511073306263934370106308, 7.33324318063674385476533728681, 7.80263293308365736895663484499, 8.809636851381838742343708043772, 9.179168606874644148307493889177, 9.946329193947377067355622730604

Graph of the ZZ-function along the critical line