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Properties

Label 2-363-11.4-c1-0-1
Degree 22
Conductor 363363
Sign 0.659−0.751i0.659 - 0.751i
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  + (−0.640 − 0.465i)2-s + (0.309 − 0.951i)3-s + (−0.424 − 1.30i)4-s + (−2.72 + 1.98i)5-s + (−0.640 + 0.465i)6-s + (0.780 + 2.40i)7-s + (−0.825 + 2.54i)8-s + (−0.809 − 0.587i)9-s + 2.67·10-s − 1.37·12-s + (4.72 + 3.43i)13-s + (0.618 − 1.90i)14-s + (1.04 + 3.20i)15-s + (−0.507 + 0.368i)16-s + (−2.16 + 1.57i)17-s + (0.244 + 0.753i)18-s + ⋯
L(s)  = 1  + (−0.453 − 0.329i)2-s + (0.178 − 0.549i)3-s + (−0.212 − 0.652i)4-s + (−1.22 + 0.886i)5-s + (−0.261 + 0.190i)6-s + (0.294 + 0.907i)7-s + (−0.291 + 0.898i)8-s + (−0.269 − 0.195i)9-s + 0.844·10-s − 0.396·12-s + (1.31 + 0.952i)13-s + (0.165 − 0.508i)14-s + (0.269 + 0.828i)15-s + (−0.126 + 0.0922i)16-s + (−0.524 + 0.380i)17-s + (0.0577 + 0.177i)18-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=((0.659−0.751i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+1/2) L(s)=((0.659−0.751i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: 0.659−0.751i0.659 - 0.751i
Analytic conductor: 2.898562.89856
Root analytic conductor: 1.702511.70251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ363(202,⋅)\chi_{363} (202, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 363, ( :1/2), 0.659−0.751i)(2,\ 363,\ (\ :1/2),\ 0.659 - 0.751i)

Particular Values

L(1)L(1) ≈\approx 0.621236+0.281471i0.621236 + 0.281471i
L(12)L(\frac12) ≈\approx 0.621236+0.281471i0.621236 + 0.281471i
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+(0.640+0.465i)T+(0.618+1.90i)T2 1 + (0.640 + 0.465i)T + (0.618 + 1.90i)T^{2}
5 1+(2.72−1.98i)T+(1.54−4.75i)T2 1 + (2.72 - 1.98i)T + (1.54 - 4.75i)T^{2}
7 1+(−0.780−2.40i)T+(−5.66+4.11i)T2 1 + (-0.780 - 2.40i)T + (-5.66 + 4.11i)T^{2}
13 1+(−4.72−3.43i)T+(4.01+12.3i)T2 1 + (-4.72 - 3.43i)T + (4.01 + 12.3i)T^{2}
17 1+(2.16−1.57i)T+(5.25−16.1i)T2 1 + (2.16 - 1.57i)T + (5.25 - 16.1i)T^{2}
19 1+(0.290−0.893i)T+(−15.3−11.1i)T2 1 + (0.290 - 0.893i)T + (-15.3 - 11.1i)T^{2}
23 1−2T+23T2 1 - 2T + 23T^{2}
29 1+(0.244+0.753i)T+(−23.4+17.0i)T2 1 + (0.244 + 0.753i)T + (-23.4 + 17.0i)T^{2}
31 1+(1.31+0.956i)T+(9.57+29.4i)T2 1 + (1.31 + 0.956i)T + (9.57 + 29.4i)T^{2}
37 1+(−1.54−4.75i)T+(−29.9+21.7i)T2 1 + (-1.54 - 4.75i)T + (-29.9 + 21.7i)T^{2}
41 1+(3.36−10.3i)T+(−33.1−24.0i)T2 1 + (3.36 - 10.3i)T + (-33.1 - 24.0i)T^{2}
43 1+6.63T+43T2 1 + 6.63T + 43T^{2}
47 1+(3.93−12.1i)T+(−38.0−27.6i)T2 1 + (3.93 - 12.1i)T + (-38.0 - 27.6i)T^{2}
53 1+(−3.33−2.41i)T+(16.3+50.4i)T2 1 + (-3.33 - 2.41i)T + (16.3 + 50.4i)T^{2}
59 1+(1.85+5.70i)T+(−47.7+34.6i)T2 1 + (1.85 + 5.70i)T + (-47.7 + 34.6i)T^{2}
61 1+(4.84−3.51i)T+(18.8−58.0i)T2 1 + (4.84 - 3.51i)T + (18.8 - 58.0i)T^{2}
67 1+1.11T+67T2 1 + 1.11T + 67T^{2}
71 1+(−8.69+6.31i)T+(21.9−67.5i)T2 1 + (-8.69 + 6.31i)T + (21.9 - 67.5i)T^{2}
73 1+(−2.82−8.70i)T+(−59.0+42.9i)T2 1 + (-2.82 - 8.70i)T + (-59.0 + 42.9i)T^{2}
79 1+(3.32+2.41i)T+(24.4+75.1i)T2 1 + (3.32 + 2.41i)T + (24.4 + 75.1i)T^{2}
83 1+(−1.52+1.10i)T+(25.6−78.9i)T2 1 + (-1.52 + 1.10i)T + (25.6 - 78.9i)T^{2}
89 1+0.627T+89T2 1 + 0.627T + 89T^{2}
97 1+(8.48+6.16i)T+(29.9+92.2i)T2 1 + (8.48 + 6.16i)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

−11.22385886147795236156437589547, −11.09042336375595841971720268031, −9.653240433288279454726329650336, −8.613001088628259718005862859392, −8.127566293261535037359616237281, −6.76255809709868602627194491776, −6.00415064640630737572942878594, −4.46644554339049063542544908055, −3.06390081618628394003676338688, −1.65280666325032177833019229412, 0.56121466692666824421319245914, 3.50166125042012130682780636727, 4.05059824164538791965083274872, 5.14190955941689848332265825502, 6.92616860817022800516721995721, 7.81807446335303163409949972739, 8.494919337026937123032233108265, 9.056251475346930238196586507052, 10.40881338012991056173598471451, 11.22580558067732439712456195562

Graph of the ZZ-function along the critical line