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Properties

Label 2-31e2-31.4-c1-0-49
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  + (−0.213 − 0.655i)2-s + (0.278 − 0.858i)3-s + (1.23 − 0.896i)4-s + 3.70·5-s − 0.622·6-s + (−0.617 + 0.448i)7-s + (−1.96 − 1.42i)8-s + (1.76 + 1.28i)9-s + (−0.789 − 2.43i)10-s + (−3.32 + 2.41i)11-s + (−0.425 − 1.30i)12-s + (0.899 − 2.76i)13-s + (0.425 + 0.309i)14-s + (1.03 − 3.18i)15-s + (0.424 − 1.30i)16-s + (1.05 + 0.769i)17-s + ⋯
L(s)  = 1  + (−0.150 − 0.463i)2-s + (0.160 − 0.495i)3-s + (0.616 − 0.448i)4-s + 1.65·5-s − 0.254·6-s + (−0.233 + 0.169i)7-s + (−0.695 − 0.505i)8-s + (0.589 + 0.428i)9-s + (−0.249 − 0.768i)10-s + (−1.00 + 0.729i)11-s + (−0.122 − 0.377i)12-s + (0.249 − 0.767i)13-s + (0.113 + 0.0826i)14-s + (0.266 − 0.821i)15-s + (0.106 − 0.326i)16-s + (0.256 + 0.186i)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 1.68571−1.59937i1.68571 - 1.59937i
L(12)L(\frac12) ≈\approx 1.68571−1.59937i1.68571 - 1.59937i
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+(0.213+0.655i)T+(−1.61+1.17i)T2 1 + (0.213 + 0.655i)T + (-1.61 + 1.17i)T^{2}
3 1+(−0.278+0.858i)T+(−2.42−1.76i)T2 1 + (-0.278 + 0.858i)T + (-2.42 - 1.76i)T^{2}
5 1−3.70T+5T2 1 - 3.70T + 5T^{2}
7 1+(0.617−0.448i)T+(2.16−6.65i)T2 1 + (0.617 - 0.448i)T + (2.16 - 6.65i)T^{2}
11 1+(3.32−2.41i)T+(3.39−10.4i)T2 1 + (3.32 - 2.41i)T + (3.39 - 10.4i)T^{2}
13 1+(−0.899+2.76i)T+(−10.5−7.64i)T2 1 + (-0.899 + 2.76i)T + (-10.5 - 7.64i)T^{2}
17 1+(−1.05−0.769i)T+(5.25+16.1i)T2 1 + (-1.05 - 0.769i)T + (5.25 + 16.1i)T^{2}
19 1+(1.17+3.61i)T+(−15.3+11.1i)T2 1 + (1.17 + 3.61i)T + (-15.3 + 11.1i)T^{2}
23 1+(2.65+1.92i)T+(7.10+21.8i)T2 1 + (2.65 + 1.92i)T + (7.10 + 21.8i)T^{2}
29 1+(1.51+4.65i)T+(−23.4+17.0i)T2 1 + (1.51 + 4.65i)T + (-23.4 + 17.0i)T^{2}
37 1−10.4T+37T2 1 - 10.4T + 37T^{2}
41 1+(−0.233−0.718i)T+(−33.1+24.0i)T2 1 + (-0.233 - 0.718i)T + (-33.1 + 24.0i)T^{2}
43 1+(−2.21−6.80i)T+(−34.7+25.2i)T2 1 + (-2.21 - 6.80i)T + (-34.7 + 25.2i)T^{2}
47 1+(0.270−0.833i)T+(−38.0−27.6i)T2 1 + (0.270 - 0.833i)T + (-38.0 - 27.6i)T^{2}
53 1+(−2.89−2.10i)T+(16.3+50.4i)T2 1 + (-2.89 - 2.10i)T + (16.3 + 50.4i)T^{2}
59 1+(−0.286+0.881i)T+(−47.7−34.6i)T2 1 + (-0.286 + 0.881i)T + (-47.7 - 34.6i)T^{2}
61 1+2.31T+61T2 1 + 2.31T + 61T^{2}
67 1+2.08T+67T2 1 + 2.08T + 67T^{2}
71 1+(−6.25−4.54i)T+(21.9+67.5i)T2 1 + (-6.25 - 4.54i)T + (21.9 + 67.5i)T^{2}
73 1+(4.57−3.32i)T+(22.5−69.4i)T2 1 + (4.57 - 3.32i)T + (22.5 - 69.4i)T^{2}
79 1+(11.4+8.28i)T+(24.4+75.1i)T2 1 + (11.4 + 8.28i)T + (24.4 + 75.1i)T^{2}
83 1+(−4.35−13.4i)T+(−67.1+48.7i)T2 1 + (-4.35 - 13.4i)T + (-67.1 + 48.7i)T^{2}
89 1+(3.58−2.60i)T+(27.5−84.6i)T2 1 + (3.58 - 2.60i)T + (27.5 - 84.6i)T^{2}
97 1+(−4.26+3.10i)T+(29.9−92.2i)T2 1 + (-4.26 + 3.10i)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

−9.915757306564232011238613297360, −9.422816833548723504795657072614, −8.095233880530206128673800976436, −7.22711210637073110943246270863, −6.24645620934715100742675047151, −5.75696105920472299166796201272, −4.67558544588175958104240677240, −2.65669696452825218333135857135, −2.33626463801953113023479446943, −1.20402119077108884747539040169, 1.72099140785476511691342866512, 2.80514261730175530961083396966, 3.85026470887513976802235809849, 5.31508189739776233501043310088, 6.02290983632533836622970562626, 6.69429815590218498977403454789, 7.65836006279002550622636081016, 8.689182443546172184743006138482, 9.383091852239930357484920336707, 10.16228516421618274233802459782

Graph of the ZZ-function along the critical line