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Properties

Label 2-363-11.3-c1-0-10
Degree 22
Conductor 363363
Sign 0.0694+0.997i0.0694 + 0.997i
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 + 1.31i)2-s + (0.309 + 0.951i)3-s + (0.927 − 2.85i)4-s + (−1.61 − 1.17i)5-s + (−1.80 − 1.31i)6-s + (−1.38 + 4.25i)7-s + (0.690 + 2.12i)8-s + (−0.809 + 0.587i)9-s + 4.47·10-s + 2.99·12-s + (−3.09 − 9.51i)14-s + (0.618 − 1.90i)15-s + (0.809 + 0.587i)16-s + (−3.61 − 2.62i)17-s + (0.690 − 2.12i)18-s + (−1.38 − 4.25i)19-s + ⋯
L(s)  = 1  + (−1.27 + 0.929i)2-s + (0.178 + 0.549i)3-s + (0.463 − 1.42i)4-s + (−0.723 − 0.525i)5-s + (−0.738 − 0.536i)6-s + (−0.522 + 1.60i)7-s + (0.244 + 0.751i)8-s + (−0.269 + 0.195i)9-s + 1.41·10-s + 0.866·12-s + (−0.825 − 2.54i)14-s + (0.159 − 0.491i)15-s + (0.202 + 0.146i)16-s + (−0.877 − 0.637i)17-s + (0.162 − 0.501i)18-s + (−0.317 − 0.975i)19-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=((0.0694+0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+1/2) L(s)=((0.0694+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: 0.0694+0.997i0.0694 + 0.997i
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: 11
Selberg data: (2, 363, ( :1/2), 0.0694+0.997i)(2,\ 363,\ (\ :1/2),\ 0.0694 + 0.997i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+(1.80−1.31i)T+(0.618−1.90i)T2 1 + (1.80 - 1.31i)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+(1.38−4.25i)T+(−5.66−4.11i)T2 1 + (1.38 - 4.25i)T + (-5.66 - 4.11i)T^{2}
13 1+(4.01−12.3i)T2 1 + (4.01 - 12.3i)T^{2}
17 1+(3.61+2.62i)T+(5.25+16.1i)T2 1 + (3.61 + 2.62i)T + (5.25 + 16.1i)T^{2}
19 1+(1.38+4.25i)T+(−15.3+11.1i)T2 1 + (1.38 + 4.25i)T + (-15.3 + 11.1i)T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+(−1.38+4.25i)T+(−23.4−17.0i)T2 1 + (-1.38 + 4.25i)T + (-23.4 - 17.0i)T^{2}
31 1+(9.57−29.4i)T2 1 + (9.57 - 29.4i)T^{2}
37 1+(−0.618+1.90i)T+(−29.9−21.7i)T2 1 + (-0.618 + 1.90i)T + (-29.9 - 21.7i)T^{2}
41 1+(1.38+4.25i)T+(−33.1+24.0i)T2 1 + (1.38 + 4.25i)T + (-33.1 + 24.0i)T^{2}
43 1−4.47T+43T2 1 - 4.47T + 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+(−47.7−34.6i)T2 1 + (-47.7 - 34.6i)T^{2}
61 1+(7.23+5.25i)T+(18.8+58.0i)T2 1 + (7.23 + 5.25i)T + (18.8 + 58.0i)T^{2}
67 1+12T+67T2 1 + 12T + 67T^{2}
71 1+(−6.47−4.70i)T+(21.9+67.5i)T2 1 + (-6.47 - 4.70i)T + (21.9 + 67.5i)T^{2}
73 1+(2.76−8.50i)T+(−59.0−42.9i)T2 1 + (2.76 - 8.50i)T + (-59.0 - 42.9i)T^{2}
79 1+(10.8−7.88i)T+(24.4−75.1i)T2 1 + (10.8 - 7.88i)T + (24.4 - 75.1i)T^{2}
83 1+(−7.23−5.25i)T+(25.6+78.9i)T2 1 + (-7.23 - 5.25i)T + (25.6 + 78.9i)T^{2}
89 1+14T+89T2 1 + 14T + 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

−11.00816007690232061038191523261, −9.769634080470012183104372079288, −9.095155075075594348895604095924, −8.619710990171056090780648214955, −7.76125293115082883441175143672, −6.53554551506523268766400769498, −5.63493262137862259144867806582, −4.34907605974048514928284053301, −2.57095219232293235532905053543, 0, 1.59237093743335086277336395191, 3.18757331967181542271751575697, 4.06724442962360687649635614436, 6.37427797255033564209703097513, 7.36742244047383441975759703259, 7.919997173281999995308655437635, 8.911596981415651533460826170864, 10.11311115078579586624472779476, 10.57406753056479389385192318086

Graph of the ZZ-function along the critical line