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Properties

Label 2-825-11.3-c1-0-36
Degree 22
Conductor 825825
Sign −0.394+0.918i-0.394 + 0.918i
Analytic cond. 6.587656.58765
Root an. cond. 2.566642.56664
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  + (2.06 − 1.49i)2-s + (0.309 + 0.951i)3-s + (1.39 − 4.28i)4-s + (2.06 + 1.49i)6-s + (1.44 − 4.45i)7-s + (−1.97 − 6.09i)8-s + (−0.809 + 0.587i)9-s + (−2.54 − 2.12i)11-s + 4.51·12-s + (−2.89 + 2.10i)13-s + (−3.69 − 11.3i)14-s + (−5.92 − 4.30i)16-s + (1.02 + 0.741i)17-s + (−0.788 + 2.42i)18-s + (2.18 + 6.71i)19-s + ⋯
L(s)  = 1  + (1.45 − 1.06i)2-s + (0.178 + 0.549i)3-s + (0.696 − 2.14i)4-s + (0.842 + 0.612i)6-s + (0.546 − 1.68i)7-s + (−0.699 − 2.15i)8-s + (−0.269 + 0.195i)9-s + (−0.767 − 0.641i)11-s + 1.30·12-s + (−0.802 + 0.583i)13-s + (−0.986 − 3.03i)14-s + (−1.48 − 1.07i)16-s + (0.247 + 0.179i)17-s + (−0.185 + 0.571i)18-s + (0.500 + 1.54i)19-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=((−0.394+0.918i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s+1/2) L(s)=((−0.394+0.918i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: −0.394+0.918i-0.394 + 0.918i
Analytic conductor: 6.587656.58765
Root analytic conductor: 2.566642.56664
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ825(751,⋅)\chi_{825} (751, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 825, ( :1/2), −0.394+0.918i)(2,\ 825,\ (\ :1/2),\ -0.394 + 0.918i)

Particular Values

L(1)L(1) ≈\approx 1.96711−2.98575i1.96711 - 2.98575i
L(12)L(\frac12) ≈\approx 1.96711−2.98575i1.96711 - 2.98575i
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)
bad3 1+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
5 1 1
11 1+(2.54+2.12i)T 1 + (2.54 + 2.12i)T
good2 1+(−2.06+1.49i)T+(0.618−1.90i)T2 1 + (-2.06 + 1.49i)T + (0.618 - 1.90i)T^{2}
7 1+(−1.44+4.45i)T+(−5.66−4.11i)T2 1 + (-1.44 + 4.45i)T + (-5.66 - 4.11i)T^{2}
13 1+(2.89−2.10i)T+(4.01−12.3i)T2 1 + (2.89 - 2.10i)T + (4.01 - 12.3i)T^{2}
17 1+(−1.02−0.741i)T+(5.25+16.1i)T2 1 + (-1.02 - 0.741i)T + (5.25 + 16.1i)T^{2}
19 1+(−2.18−6.71i)T+(−15.3+11.1i)T2 1 + (-2.18 - 6.71i)T + (-15.3 + 11.1i)T^{2}
23 1−3.08T+23T2 1 - 3.08T + 23T^{2}
29 1+(−1.91+5.89i)T+(−23.4−17.0i)T2 1 + (-1.91 + 5.89i)T + (-23.4 - 17.0i)T^{2}
31 1+(−0.0537+0.0390i)T+(9.57−29.4i)T2 1 + (-0.0537 + 0.0390i)T + (9.57 - 29.4i)T^{2}
37 1+(1.22−3.76i)T+(−29.9−21.7i)T2 1 + (1.22 - 3.76i)T + (-29.9 - 21.7i)T^{2}
41 1+(−1.62−4.99i)T+(−33.1+24.0i)T2 1 + (-1.62 - 4.99i)T + (-33.1 + 24.0i)T^{2}
43 1−9.77T+43T2 1 - 9.77T + 43T^{2}
47 1+(−1.67−5.14i)T+(−38.0+27.6i)T2 1 + (-1.67 - 5.14i)T + (-38.0 + 27.6i)T^{2}
53 1+(−10.6+7.73i)T+(16.3−50.4i)T2 1 + (-10.6 + 7.73i)T + (16.3 - 50.4i)T^{2}
59 1+(0.163−0.504i)T+(−47.7−34.6i)T2 1 + (0.163 - 0.504i)T + (-47.7 - 34.6i)T^{2}
61 1+(4.96+3.60i)T+(18.8+58.0i)T2 1 + (4.96 + 3.60i)T + (18.8 + 58.0i)T^{2}
67 1−3.10T+67T2 1 - 3.10T + 67T^{2}
71 1+(6.73+4.89i)T+(21.9+67.5i)T2 1 + (6.73 + 4.89i)T + (21.9 + 67.5i)T^{2}
73 1+(5.15−15.8i)T+(−59.0−42.9i)T2 1 + (5.15 - 15.8i)T + (-59.0 - 42.9i)T^{2}
79 1+(6.10−4.43i)T+(24.4−75.1i)T2 1 + (6.10 - 4.43i)T + (24.4 - 75.1i)T^{2}
83 1+(−3.90−2.83i)T+(25.6+78.9i)T2 1 + (-3.90 - 2.83i)T + (25.6 + 78.9i)T^{2}
89 1+6.05T+89T2 1 + 6.05T + 89T^{2}
97 1+(2.46−1.78i)T+(29.9−92.2i)T2 1 + (2.46 - 1.78i)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

−10.19263148441652013579170267987, −9.795563582664646071608927322196, −8.166936941790772380694287532326, −7.30134431506059343314829401534, −6.02191989988917938816207639683, −5.09013800819168683785116099170, −4.30145580170787524558322527961, −3.68734174455607948897651815635, −2.61461284062223799337413220518, −1.18172891194111928982139578493, 2.44923188731380357378814177831, 2.92833375736992886444108879329, 4.65624627767906692533883878429, 5.30243425989170602074767360989, 5.81439811280273294111660844220, 7.17110806295462215082306810796, 7.42711857848070311709757064866, 8.537507550940299132562811703560, 9.207249036333172858807331846070, 10.78948724293300551954534589049

Graph of the ZZ-function along the critical line