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Properties

Label 2-363-11.3-c1-0-8
Degree 22
Conductor 363363
Sign 0.995+0.0913i0.995 + 0.0913i
Analytic cond. 2.898562.89856
Root an. cond. 1.702511.70251
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  + (−0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (−0.309 + 0.951i)4-s + (1.61 + 1.17i)5-s + (0.809 + 0.587i)6-s + (1.23 − 3.80i)7-s + (−0.927 − 2.85i)8-s + (−0.809 + 0.587i)9-s − 2·10-s + 0.999·12-s + (1.61 − 1.17i)13-s + (1.23 + 3.80i)14-s + (0.618 − 1.90i)15-s + (0.809 + 0.587i)16-s + (1.61 + 1.17i)17-s + (0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.178 − 0.549i)3-s + (−0.154 + 0.475i)4-s + (0.723 + 0.525i)5-s + (0.330 + 0.239i)6-s + (0.467 − 1.43i)7-s + (−0.327 − 1.00i)8-s + (−0.269 + 0.195i)9-s − 0.632·10-s + 0.288·12-s + (0.448 − 0.326i)13-s + (0.330 + 1.01i)14-s + (0.159 − 0.491i)15-s + (0.202 + 0.146i)16-s + (0.392 + 0.285i)17-s + (0.0728 − 0.224i)18-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=((0.995+0.0913i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0913i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+1/2) L(s)=((0.995+0.0913i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: 0.995+0.0913i0.995 + 0.0913i
Analytic conductor: 2.898562.89856
Root analytic conductor: 1.702511.70251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ363(124,⋅)\chi_{363} (124, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 363, ( :1/2), 0.995+0.0913i)(2,\ 363,\ (\ :1/2),\ 0.995 + 0.0913i)

Particular Values

L(1)L(1) ≈\approx 1.05846−0.0484686i1.05846 - 0.0484686i
L(12)L(\frac12) ≈\approx 1.05846−0.0484686i1.05846 - 0.0484686i
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
11 1 1
good2 1+(0.809−0.587i)T+(0.618−1.90i)T2 1 + (0.809 - 0.587i)T + (0.618 - 1.90i)T^{2}
5 1+(−1.61−1.17i)T+(1.54+4.75i)T2 1 + (-1.61 - 1.17i)T + (1.54 + 4.75i)T^{2}
7 1+(−1.23+3.80i)T+(−5.66−4.11i)T2 1 + (-1.23 + 3.80i)T + (-5.66 - 4.11i)T^{2}
13 1+(−1.61+1.17i)T+(4.01−12.3i)T2 1 + (-1.61 + 1.17i)T + (4.01 - 12.3i)T^{2}
17 1+(−1.61−1.17i)T+(5.25+16.1i)T2 1 + (-1.61 - 1.17i)T + (5.25 + 16.1i)T^{2}
19 1+(−15.3+11.1i)T2 1 + (-15.3 + 11.1i)T^{2}
23 1−8T+23T2 1 - 8T + 23T^{2}
29 1+(1.85−5.70i)T+(−23.4−17.0i)T2 1 + (1.85 - 5.70i)T + (-23.4 - 17.0i)T^{2}
31 1+(−6.47+4.70i)T+(9.57−29.4i)T2 1 + (-6.47 + 4.70i)T + (9.57 - 29.4i)T^{2}
37 1+(−1.85+5.70i)T+(−29.9−21.7i)T2 1 + (-1.85 + 5.70i)T + (-29.9 - 21.7i)T^{2}
41 1+(0.618+1.90i)T+(−33.1+24.0i)T2 1 + (0.618 + 1.90i)T + (-33.1 + 24.0i)T^{2}
43 1+43T2 1 + 43T^{2}
47 1+(−2.47−7.60i)T+(−38.0+27.6i)T2 1 + (-2.47 - 7.60i)T + (-38.0 + 27.6i)T^{2}
53 1+(4.85−3.52i)T+(16.3−50.4i)T2 1 + (4.85 - 3.52i)T + (16.3 - 50.4i)T^{2}
59 1+(1.23−3.80i)T+(−47.7−34.6i)T2 1 + (1.23 - 3.80i)T + (-47.7 - 34.6i)T^{2}
61 1+(4.85+3.52i)T+(18.8+58.0i)T2 1 + (4.85 + 3.52i)T + (18.8 + 58.0i)T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1+(21.9+67.5i)T2 1 + (21.9 + 67.5i)T^{2}
73 1+(4.32−13.3i)T+(−59.0−42.9i)T2 1 + (4.32 - 13.3i)T + (-59.0 - 42.9i)T^{2}
79 1+(−3.23+2.35i)T+(24.4−75.1i)T2 1 + (-3.23 + 2.35i)T + (24.4 - 75.1i)T^{2}
83 1+(9.70+7.05i)T+(25.6+78.9i)T2 1 + (9.70 + 7.05i)T + (25.6 + 78.9i)T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+(1.61−1.17i)T+(29.9−92.2i)T2 1 + (1.61 - 1.17i)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

−11.14035074097355378837046640489, −10.51848162273505856752290011408, −9.532947179135651459871728145427, −8.428035839641687917853336516087, −7.51155274978116669969181040823, −6.95942308197040727936427706270, −5.94311890327935492725520166309, −4.37718058628109478588777029219, −3.05472149613818657601559610566, −1.08779967835696234453711612737, 1.45331848047328978567482974789, 2.76978359856214426281299607856, 4.84801067984529428985842469974, 5.41176829238472466554664166835, 6.31574932757034554537252582885, 8.246268470036361314252354269464, 9.012676679030885163482067928499, 9.460846099457401586346010017683, 10.38832560941731692992475797034, 11.40856045521685175147980469806

Graph of the ZZ-function along the critical line