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Properties

Label 2-363-11.4-c1-0-13
Degree 22
Conductor 363363
Sign 0.927+0.374i0.927 + 0.374i
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  + (1.30 + 0.951i)2-s + (0.309 − 0.951i)3-s + (0.190 + 0.587i)4-s + (0.309 − 0.224i)5-s + (1.30 − 0.951i)6-s + (−0.927 − 2.85i)7-s + (0.690 − 2.12i)8-s + (−0.809 − 0.587i)9-s + 0.618·10-s + 0.618·12-s + (5.04 + 3.66i)13-s + (1.5 − 4.61i)14-s + (−0.118 − 0.363i)15-s + (3.92 − 2.85i)16-s + (−0.5 + 0.363i)17-s + (−0.499 − 1.53i)18-s + ⋯
L(s)  = 1  + (0.925 + 0.672i)2-s + (0.178 − 0.549i)3-s + (0.0954 + 0.293i)4-s + (0.138 − 0.100i)5-s + (0.534 − 0.388i)6-s + (−0.350 − 1.07i)7-s + (0.244 − 0.751i)8-s + (−0.269 − 0.195i)9-s + 0.195·10-s + 0.178·12-s + (1.39 + 1.01i)13-s + (0.400 − 1.23i)14-s + (−0.0304 − 0.0937i)15-s + (0.981 − 0.713i)16-s + (−0.121 + 0.0881i)17-s + (−0.117 − 0.362i)18-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=((0.927+0.374i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+1/2) L(s)=((0.927+0.374i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: 0.927+0.374i0.927 + 0.374i
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), 0.927+0.374i)(2,\ 363,\ (\ :1/2),\ 0.927 + 0.374i)

Particular Values

L(1)L(1) ≈\approx 2.18789−0.424966i2.18789 - 0.424966i
L(12)L(\frac12) ≈\approx 2.18789−0.424966i2.18789 - 0.424966i
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+(−1.30−0.951i)T+(0.618+1.90i)T2 1 + (-1.30 - 0.951i)T + (0.618 + 1.90i)T^{2}
5 1+(−0.309+0.224i)T+(1.54−4.75i)T2 1 + (-0.309 + 0.224i)T + (1.54 - 4.75i)T^{2}
7 1+(0.927+2.85i)T+(−5.66+4.11i)T2 1 + (0.927 + 2.85i)T + (-5.66 + 4.11i)T^{2}
13 1+(−5.04−3.66i)T+(4.01+12.3i)T2 1 + (-5.04 - 3.66i)T + (4.01 + 12.3i)T^{2}
17 1+(0.5−0.363i)T+(5.25−16.1i)T2 1 + (0.5 - 0.363i)T + (5.25 - 16.1i)T^{2}
19 1+(−0.263+0.812i)T+(−15.3−11.1i)T2 1 + (-0.263 + 0.812i)T + (-15.3 - 11.1i)T^{2}
23 1+5.47T+23T2 1 + 5.47T + 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+(−3.11−2.26i)T+(9.57+29.4i)T2 1 + (-3.11 - 2.26i)T + (9.57 + 29.4i)T^{2}
37 1+(1.30+4.02i)T+(−29.9+21.7i)T2 1 + (1.30 + 4.02i)T + (-29.9 + 21.7i)T^{2}
41 1+(1.83−5.65i)T+(−33.1−24.0i)T2 1 + (1.83 - 5.65i)T + (-33.1 - 24.0i)T^{2}
43 1+1.76T+43T2 1 + 1.76T + 43T^{2}
47 1+(0.190−0.587i)T+(−38.0−27.6i)T2 1 + (0.190 - 0.587i)T + (-38.0 - 27.6i)T^{2}
53 1+(−5.97−4.33i)T+(16.3+50.4i)T2 1 + (-5.97 - 4.33i)T + (16.3 + 50.4i)T^{2}
59 1+(1.64+5.06i)T+(−47.7+34.6i)T2 1 + (1.64 + 5.06i)T + (-47.7 + 34.6i)T^{2}
61 1+(−0.927+0.673i)T+(18.8−58.0i)T2 1 + (-0.927 + 0.673i)T + (18.8 - 58.0i)T^{2}
67 1−10.5T+67T2 1 - 10.5T + 67T^{2}
71 1+(11.7−8.55i)T+(21.9−67.5i)T2 1 + (11.7 - 8.55i)T + (21.9 - 67.5i)T^{2}
73 1+(0.381+1.17i)T+(−59.0+42.9i)T2 1 + (0.381 + 1.17i)T + (-59.0 + 42.9i)T^{2}
79 1+(−0.427−0.310i)T+(24.4+75.1i)T2 1 + (-0.427 - 0.310i)T + (24.4 + 75.1i)T^{2}
83 1+(10.2−7.46i)T+(25.6−78.9i)T2 1 + (10.2 - 7.46i)T + (25.6 - 78.9i)T^{2}
89 1−9.47T+89T2 1 - 9.47T + 89T^{2}
97 1+(12.1+8.83i)T+(29.9+92.2i)T2 1 + (12.1 + 8.83i)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

−11.53153967875107988974687381461, −10.50175440600152185558390426096, −9.502511511838861163313282369242, −8.359442629740444964680472338572, −7.16174617959020833457234934033, −6.59499183901965007070164906760, −5.69529136386931716985065287513, −4.32439820119788798275205486366, −3.53235334792978867159417635623, −1.35730264739982931749737324215, 2.29116944987890974546669924597, 3.28885919867012159462202789842, 4.24391109747656867894977990437, 5.53715165773940391836273667989, 6.13425150205134231176233639154, 8.101826672997573657269312259242, 8.629188344538009554208504873035, 9.925672497820429654364093572375, 10.67292136303796175608522307314, 11.73150049139990531066128980259

Graph of the ZZ-function along the critical line