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Properties

Label 2-231-11.4-c1-0-7
Degree 22
Conductor 231231
Sign 0.0694−0.997i0.0694 - 0.997i
Analytic cond. 1.844541.84454
Root an. cond. 1.358141.35814
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  + (1.93 + 1.40i)2-s + (−0.309 + 0.951i)3-s + (1.14 + 3.53i)4-s + (2.09 − 1.52i)5-s + (−1.93 + 1.40i)6-s + (−0.309 − 0.951i)7-s + (−1.26 + 3.89i)8-s + (−0.809 − 0.587i)9-s + 6.19·10-s + (−2.19 + 2.48i)11-s − 3.71·12-s + (−5.48 − 3.98i)13-s + (0.738 − 2.27i)14-s + (0.800 + 2.46i)15-s + (−1.91 + 1.38i)16-s + (2.54 − 1.84i)17-s + ⋯
L(s)  = 1  + (1.36 + 0.993i)2-s + (−0.178 + 0.549i)3-s + (0.573 + 1.76i)4-s + (0.937 − 0.680i)5-s + (−0.789 + 0.573i)6-s + (−0.116 − 0.359i)7-s + (−0.447 + 1.37i)8-s + (−0.269 − 0.195i)9-s + 1.95·10-s + (−0.660 + 0.750i)11-s − 1.07·12-s + (−1.52 − 1.10i)13-s + (0.197 − 0.607i)14-s + (0.206 + 0.635i)15-s + (−0.477 + 0.347i)16-s + (0.617 − 0.448i)17-s + ⋯

Functional equation

Λ(s)=(231s/2 ΓC(s) L(s)=((0.0694−0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(231s/2 ΓC(s+1/2) L(s)=((0.0694−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{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: 231231    =    3⋅7⋅113 \cdot 7 \cdot 11
Sign: 0.0694−0.997i0.0694 - 0.997i
Analytic conductor: 1.844541.84454
Root analytic conductor: 1.358141.35814
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ231(169,⋅)\chi_{231} (169, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 231, ( :1/2), 0.0694−0.997i)(2,\ 231,\ (\ :1/2),\ 0.0694 - 0.997i)

Particular Values

L(1)L(1) ≈\approx 1.76470+1.64607i1.76470 + 1.64607i
L(12)L(\frac12) ≈\approx 1.76470+1.64607i1.76470 + 1.64607i
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
7 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
11 1+(2.19−2.48i)T 1 + (2.19 - 2.48i)T
good2 1+(−1.93−1.40i)T+(0.618+1.90i)T2 1 + (-1.93 - 1.40i)T + (0.618 + 1.90i)T^{2}
5 1+(−2.09+1.52i)T+(1.54−4.75i)T2 1 + (-2.09 + 1.52i)T + (1.54 - 4.75i)T^{2}
13 1+(5.48+3.98i)T+(4.01+12.3i)T2 1 + (5.48 + 3.98i)T + (4.01 + 12.3i)T^{2}
17 1+(−2.54+1.84i)T+(5.25−16.1i)T2 1 + (-2.54 + 1.84i)T + (5.25 - 16.1i)T^{2}
19 1+(0.323−0.994i)T+(−15.3−11.1i)T2 1 + (0.323 - 0.994i)T + (-15.3 - 11.1i)T^{2}
23 1+6.52T+23T2 1 + 6.52T + 23T^{2}
29 1+(0.187+0.577i)T+(−23.4+17.0i)T2 1 + (0.187 + 0.577i)T + (-23.4 + 17.0i)T^{2}
31 1+(−7.14−5.19i)T+(9.57+29.4i)T2 1 + (-7.14 - 5.19i)T + (9.57 + 29.4i)T^{2}
37 1+(−2.76−8.50i)T+(−29.9+21.7i)T2 1 + (-2.76 - 8.50i)T + (-29.9 + 21.7i)T^{2}
41 1+(−2.68+8.27i)T+(−33.1−24.0i)T2 1 + (-2.68 + 8.27i)T + (-33.1 - 24.0i)T^{2}
43 1−4.48T+43T2 1 - 4.48T + 43T^{2}
47 1+(−1.00+3.10i)T+(−38.0−27.6i)T2 1 + (-1.00 + 3.10i)T + (-38.0 - 27.6i)T^{2}
53 1+(1.53+1.11i)T+(16.3+50.4i)T2 1 + (1.53 + 1.11i)T + (16.3 + 50.4i)T^{2}
59 1+(0.0537+0.165i)T+(−47.7+34.6i)T2 1 + (0.0537 + 0.165i)T + (-47.7 + 34.6i)T^{2}
61 1+(6.58−4.78i)T+(18.8−58.0i)T2 1 + (6.58 - 4.78i)T + (18.8 - 58.0i)T^{2}
67 1+7.33T+67T2 1 + 7.33T + 67T^{2}
71 1+(2.18−1.59i)T+(21.9−67.5i)T2 1 + (2.18 - 1.59i)T + (21.9 - 67.5i)T^{2}
73 1+(−0.182−0.561i)T+(−59.0+42.9i)T2 1 + (-0.182 - 0.561i)T + (-59.0 + 42.9i)T^{2}
79 1+(−7.93−5.76i)T+(24.4+75.1i)T2 1 + (-7.93 - 5.76i)T + (24.4 + 75.1i)T^{2}
83 1+(7.07−5.14i)T+(25.6−78.9i)T2 1 + (7.07 - 5.14i)T + (25.6 - 78.9i)T^{2}
89 1+11.6T+89T2 1 + 11.6T + 89T^{2}
97 1+(9.58+6.96i)T+(29.9+92.2i)T2 1 + (9.58 + 6.96i)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

−12.52479198209999695933161606525, −12.14738900534197005417811173208, −10.19934800962446046832581985443, −9.800666388359149410125039333038, −8.104348547723287483056875811544, −7.20370829345358660263463881052, −5.85655699414096987888690982779, −5.18651653818261368267183667592, −4.43108838161651007602865742567, −2.80957941502296216669868667512, 2.09664307140790738987860306724, 2.79867750132441243865181102732, 4.47457503547231273357968849532, 5.77053156356872025649091120115, 6.27728051623974454152427226003, 7.76651433266631817888500822588, 9.564886732528583668366910094351, 10.33895500937191667867869773257, 11.31842862709336214714387439788, 12.11628468005586752217530211780

Graph of the ZZ-function along the critical line