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Properties

Label 2-231-11.4-c1-0-6
Degree 22
Conductor 231231
Sign 0.116+0.993i0.116 + 0.993i
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  + (−0.5 − 0.363i)2-s + (−0.309 + 0.951i)3-s + (−0.5 − 1.53i)4-s + (1.30 − 0.951i)5-s + (0.5 − 0.363i)6-s + (−0.309 − 0.951i)7-s + (−0.690 + 2.12i)8-s + (−0.809 − 0.587i)9-s − 10-s + (1.69 − 2.85i)11-s + 1.61·12-s + (−3.42 − 2.48i)13-s + (−0.190 + 0.587i)14-s + (0.499 + 1.53i)15-s + (−1.49 + 1.08i)16-s + (2.80 − 2.04i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.256i)2-s + (−0.178 + 0.549i)3-s + (−0.250 − 0.769i)4-s + (0.585 − 0.425i)5-s + (0.204 − 0.148i)6-s + (−0.116 − 0.359i)7-s + (−0.244 + 0.751i)8-s + (−0.269 − 0.195i)9-s − 0.316·10-s + (0.509 − 0.860i)11-s + 0.467·12-s + (−0.950 − 0.690i)13-s + (−0.0510 + 0.157i)14-s + (0.129 + 0.397i)15-s + (−0.374 + 0.272i)16-s + (0.681 − 0.494i)17-s + ⋯

Functional equation

Λ(s)=(231s/2 ΓC(s) L(s)=((0.116+0.993i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(231s/2 ΓC(s+1/2) L(s)=((0.116+0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 231231    =    3⋅7⋅113 \cdot 7 \cdot 11
Sign: 0.116+0.993i0.116 + 0.993i
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), 0.116+0.993i)(2,\ 231,\ (\ :1/2),\ 0.116 + 0.993i)

Particular Values

L(1)L(1) ≈\approx 0.685093−0.609235i0.685093 - 0.609235i
L(12)L(\frac12) ≈\approx 0.685093−0.609235i0.685093 - 0.609235i
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+(−1.69+2.85i)T 1 + (-1.69 + 2.85i)T
good2 1+(0.5+0.363i)T+(0.618+1.90i)T2 1 + (0.5 + 0.363i)T + (0.618 + 1.90i)T^{2}
5 1+(−1.30+0.951i)T+(1.54−4.75i)T2 1 + (-1.30 + 0.951i)T + (1.54 - 4.75i)T^{2}
13 1+(3.42+2.48i)T+(4.01+12.3i)T2 1 + (3.42 + 2.48i)T + (4.01 + 12.3i)T^{2}
17 1+(−2.80+2.04i)T+(5.25−16.1i)T2 1 + (-2.80 + 2.04i)T + (5.25 - 16.1i)T^{2}
19 1+(−2+6.15i)T+(−15.3−11.1i)T2 1 + (-2 + 6.15i)T + (-15.3 - 11.1i)T^{2}
23 1−5.70T+23T2 1 - 5.70T + 23T^{2}
29 1+(0.0729+0.224i)T+(−23.4+17.0i)T2 1 + (0.0729 + 0.224i)T + (-23.4 + 17.0i)T^{2}
31 1+(2.42+1.76i)T+(9.57+29.4i)T2 1 + (2.42 + 1.76i)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+(3.04−9.37i)T+(−33.1−24.0i)T2 1 + (3.04 - 9.37i)T + (-33.1 - 24.0i)T^{2}
43 1+11.4T+43T2 1 + 11.4T + 43T^{2}
47 1+(1.59−4.89i)T+(−38.0−27.6i)T2 1 + (1.59 - 4.89i)T + (-38.0 - 27.6i)T^{2}
53 1+(−10.1−7.38i)T+(16.3+50.4i)T2 1 + (-10.1 - 7.38i)T + (16.3 + 50.4i)T^{2}
59 1+(1.42+4.39i)T+(−47.7+34.6i)T2 1 + (1.42 + 4.39i)T + (-47.7 + 34.6i)T^{2}
61 1+(−6.42+4.66i)T+(18.8−58.0i)T2 1 + (-6.42 + 4.66i)T + (18.8 - 58.0i)T^{2}
67 1−13.2T+67T2 1 - 13.2T + 67T^{2}
71 1+(−3.66+2.66i)T+(21.9−67.5i)T2 1 + (-3.66 + 2.66i)T + (21.9 - 67.5i)T^{2}
73 1+(−2.28−7.02i)T+(−59.0+42.9i)T2 1 + (-2.28 - 7.02i)T + (-59.0 + 42.9i)T^{2}
79 1+(−4.92−3.57i)T+(24.4+75.1i)T2 1 + (-4.92 - 3.57i)T + (24.4 + 75.1i)T^{2}
83 1+(6.85−4.97i)T+(25.6−78.9i)T2 1 + (6.85 - 4.97i)T + (25.6 - 78.9i)T^{2}
89 1−1.32T+89T2 1 - 1.32T + 89T^{2}
97 1+(−4.28−3.11i)T+(29.9+92.2i)T2 1 + (-4.28 - 3.11i)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

−11.59345535780682501524068540869, −10.95993559539669512538554486855, −9.727981012917189621149053394049, −9.531046756295636685489740233683, −8.369403849185161650183502390131, −6.79245754156162379511386081365, −5.46051370661874869784941089377, −4.89309863858823884918400045961, −3.00778686719332393471852774768, −0.921556623080204601596263201403, 2.09350870517998568886041453917, 3.69132467307026662966969430440, 5.32247828989768773616474982148, 6.68999949061032228405522961418, 7.28940874036668505933297838907, 8.428063946224317755795563440833, 9.486606065151950118578990536405, 10.22657074812975423694318445227, 11.88520573551656959164564504727, 12.26666157060904168190158223651

Graph of the ZZ-function along the critical line