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Properties

Label 2-693-11.4-c1-0-26
Degree 22
Conductor 693693
Sign −0.999+0.0325i-0.999 + 0.0325i
Analytic cond. 5.533635.53363
Root an. cond. 2.352362.35236
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  + (0.183 + 0.132i)2-s + (−0.602 − 1.85i)4-s + (−2.01 + 1.46i)5-s + (0.309 + 0.951i)7-s + (0.276 − 0.849i)8-s − 0.564·10-s + (2.66 − 1.97i)11-s + (−4.15 − 3.01i)13-s + (−0.0699 + 0.215i)14-s + (−2.98 + 2.17i)16-s + (−1.16 + 0.844i)17-s + (−1.87 + 5.77i)19-s + (3.93 + 2.85i)20-s + (0.750 − 0.00659i)22-s − 7.08·23-s + ⋯
L(s)  = 1  + (0.129 + 0.0940i)2-s + (−0.301 − 0.926i)4-s + (−0.902 + 0.655i)5-s + (0.116 + 0.359i)7-s + (0.0975 − 0.300i)8-s − 0.178·10-s + (0.803 − 0.594i)11-s + (−1.15 − 0.837i)13-s + (−0.0186 + 0.0574i)14-s + (−0.747 + 0.543i)16-s + (−0.282 + 0.204i)17-s + (−0.430 + 1.32i)19-s + (0.879 + 0.639i)20-s + (0.159 − 0.00140i)22-s − 1.47·23-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((−0.999+0.0325i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0325i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+1/2) L(s)=((−0.999+0.0325i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0325i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: −0.999+0.0325i-0.999 + 0.0325i
Analytic conductor: 5.533635.53363
Root analytic conductor: 2.352362.35236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ693(631,⋅)\chi_{693} (631, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 693, ( :1/2), −0.999+0.0325i)(2,\ 693,\ (\ :1/2),\ -0.999 + 0.0325i)

Particular Values

L(1)L(1) ≈\approx 0.00209558−0.128714i0.00209558 - 0.128714i
L(12)L(\frac12) ≈\approx 0.00209558−0.128714i0.00209558 - 0.128714i
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 1
7 1+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
11 1+(−2.66+1.97i)T 1 + (-2.66 + 1.97i)T
good2 1+(−0.183−0.132i)T+(0.618+1.90i)T2 1 + (-0.183 - 0.132i)T + (0.618 + 1.90i)T^{2}
5 1+(2.01−1.46i)T+(1.54−4.75i)T2 1 + (2.01 - 1.46i)T + (1.54 - 4.75i)T^{2}
13 1+(4.15+3.01i)T+(4.01+12.3i)T2 1 + (4.15 + 3.01i)T + (4.01 + 12.3i)T^{2}
17 1+(1.16−0.844i)T+(5.25−16.1i)T2 1 + (1.16 - 0.844i)T + (5.25 - 16.1i)T^{2}
19 1+(1.87−5.77i)T+(−15.3−11.1i)T2 1 + (1.87 - 5.77i)T + (-15.3 - 11.1i)T^{2}
23 1+7.08T+23T2 1 + 7.08T + 23T^{2}
29 1+(2.01+6.19i)T+(−23.4+17.0i)T2 1 + (2.01 + 6.19i)T + (-23.4 + 17.0i)T^{2}
31 1+(6.22+4.51i)T+(9.57+29.4i)T2 1 + (6.22 + 4.51i)T + (9.57 + 29.4i)T^{2}
37 1+(1.23+3.78i)T+(−29.9+21.7i)T2 1 + (1.23 + 3.78i)T + (-29.9 + 21.7i)T^{2}
41 1+(2.08−6.41i)T+(−33.1−24.0i)T2 1 + (2.08 - 6.41i)T + (-33.1 - 24.0i)T^{2}
43 1+0.802T+43T2 1 + 0.802T + 43T^{2}
47 1+(2.08−6.42i)T+(−38.0−27.6i)T2 1 + (2.08 - 6.42i)T + (-38.0 - 27.6i)T^{2}
53 1+(−5.32−3.86i)T+(16.3+50.4i)T2 1 + (-5.32 - 3.86i)T + (16.3 + 50.4i)T^{2}
59 1+(0.888+2.73i)T+(−47.7+34.6i)T2 1 + (0.888 + 2.73i)T + (-47.7 + 34.6i)T^{2}
61 1+(−0.691+0.502i)T+(18.8−58.0i)T2 1 + (-0.691 + 0.502i)T + (18.8 - 58.0i)T^{2}
67 1+1.64T+67T2 1 + 1.64T + 67T^{2}
71 1+(−3.65+2.65i)T+(21.9−67.5i)T2 1 + (-3.65 + 2.65i)T + (21.9 - 67.5i)T^{2}
73 1+(4.58+14.1i)T+(−59.0+42.9i)T2 1 + (4.58 + 14.1i)T + (-59.0 + 42.9i)T^{2}
79 1+(1.98+1.44i)T+(24.4+75.1i)T2 1 + (1.98 + 1.44i)T + (24.4 + 75.1i)T^{2}
83 1+(−1.81+1.32i)T+(25.6−78.9i)T2 1 + (-1.81 + 1.32i)T + (25.6 - 78.9i)T^{2}
89 1+1.73T+89T2 1 + 1.73T + 89T^{2}
97 1+(−9.77−7.09i)T+(29.9+92.2i)T2 1 + (-9.77 - 7.09i)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

−10.06507491731536067071844657888, −9.331807243922318950289728518739, −8.156257284375563873421612684532, −7.50381377868132041579950766258, −6.23685991520938473064640110860, −5.71693051147059081502265432133, −4.39115417714864633834381833809, −3.56607871501856589889877774450, −2.00736597968515148287647716497, −0.06211053370300269099319149531, 2.11122368624654462820966833509, 3.66950059820888842696428177278, 4.37498648445733463735319080051, 5.02174525471618439992811165167, 7.00879435627334062149179762775, 7.20344349326745189130400927345, 8.448372640695146820183419015932, 8.941143292291558286756588848566, 9.884328891776014077136354070901, 11.18004333278289520887769453830

Graph of the ZZ-function along the critical line