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Properties

Label 2-693-7.2-c1-0-29
Degree 22
Conductor 693693
Sign −0.363+0.931i-0.363 + 0.931i
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  + (0.643 − 1.11i)2-s + (0.171 + 0.296i)4-s + (1.95 − 3.39i)5-s + (−0.234 − 2.63i)7-s + 3.01·8-s + (−2.52 − 4.36i)10-s + (−0.5 − 0.866i)11-s − 3.04·13-s + (−3.08 − 1.43i)14-s + (1.59 − 2.76i)16-s + (1.98 + 3.44i)17-s + (−3.79 + 6.57i)19-s + 1.34·20-s − 1.28·22-s + (2.25 − 3.90i)23-s + ⋯
L(s)  = 1  + (0.455 − 0.788i)2-s + (0.0856 + 0.148i)4-s + (0.875 − 1.51i)5-s + (−0.0885 − 0.996i)7-s + 1.06·8-s + (−0.797 − 1.38i)10-s + (−0.150 − 0.261i)11-s − 0.843·13-s + (−0.825 − 0.383i)14-s + (0.399 − 0.692i)16-s + (0.481 + 0.834i)17-s + (−0.870 + 1.50i)19-s + 0.300·20-s − 0.274·22-s + (0.470 − 0.814i)23-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((−0.363+0.931i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+1/2) L(s)=((−0.363+0.931i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: −0.363+0.931i-0.363 + 0.931i
Analytic conductor: 5.533635.53363
Root analytic conductor: 2.352362.35236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ693(100,⋅)\chi_{693} (100, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 693, ( :1/2), −0.363+0.931i)(2,\ 693,\ (\ :1/2),\ -0.363 + 0.931i)

Particular Values

L(1)L(1) ≈\approx 1.31672−1.92629i1.31672 - 1.92629i
L(12)L(\frac12) ≈\approx 1.31672−1.92629i1.31672 - 1.92629i
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+(0.234+2.63i)T 1 + (0.234 + 2.63i)T
11 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good2 1+(−0.643+1.11i)T+(−1−1.73i)T2 1 + (-0.643 + 1.11i)T + (-1 - 1.73i)T^{2}
5 1+(−1.95+3.39i)T+(−2.5−4.33i)T2 1 + (-1.95 + 3.39i)T + (-2.5 - 4.33i)T^{2}
13 1+3.04T+13T2 1 + 3.04T + 13T^{2}
17 1+(−1.98−3.44i)T+(−8.5+14.7i)T2 1 + (-1.98 - 3.44i)T + (-8.5 + 14.7i)T^{2}
19 1+(3.79−6.57i)T+(−9.5−16.4i)T2 1 + (3.79 - 6.57i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.25+3.90i)T+(−11.5−19.9i)T2 1 + (-2.25 + 3.90i)T + (-11.5 - 19.9i)T^{2}
29 1+3.75T+29T2 1 + 3.75T + 29T^{2}
31 1+(−3.37−5.83i)T+(−15.5+26.8i)T2 1 + (-3.37 - 5.83i)T + (-15.5 + 26.8i)T^{2}
37 1+(0.171−0.296i)T+(−18.5−32.0i)T2 1 + (0.171 - 0.296i)T + (-18.5 - 32.0i)T^{2}
41 1−2.79T+41T2 1 - 2.79T + 41T^{2}
43 1−11.1T+43T2 1 - 11.1T + 43T^{2}
47 1+(−0.828+1.43i)T+(−23.5−40.7i)T2 1 + (-0.828 + 1.43i)T + (-23.5 - 40.7i)T^{2}
53 1+(6.47+11.2i)T+(−26.5+45.8i)T2 1 + (6.47 + 11.2i)T + (-26.5 + 45.8i)T^{2}
59 1+(1.83+3.17i)T+(−29.5+51.0i)T2 1 + (1.83 + 3.17i)T + (-29.5 + 51.0i)T^{2}
61 1+(−0.234+0.405i)T+(−30.5−52.8i)T2 1 + (-0.234 + 0.405i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.28−2.23i)T+(−33.5+58.0i)T2 1 + (-1.28 - 2.23i)T + (-33.5 + 58.0i)T^{2}
71 1−5.00T+71T2 1 - 5.00T + 71T^{2}
73 1+(−4.36−7.56i)T+(−36.5+63.2i)T2 1 + (-4.36 - 7.56i)T + (-36.5 + 63.2i)T^{2}
79 1+(0.359−0.623i)T+(−39.5−68.4i)T2 1 + (0.359 - 0.623i)T + (-39.5 - 68.4i)T^{2}
83 1−11.5T+83T2 1 - 11.5T + 83T^{2}
89 1+(6.17−10.6i)T+(−44.5−77.0i)T2 1 + (6.17 - 10.6i)T + (-44.5 - 77.0i)T^{2}
97 1+13.1T+97T2 1 + 13.1T + 97T^{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

−10.29130827137076573985758302154, −9.616952575274705085160702605056, −8.415071190788262680431273893222, −7.81621663085720393710227774795, −6.52569400836260047459698561555, −5.37522559812168880967676387211, −4.50770376737198839066963067656, −3.70788911283629576013094729562, −2.18272001752226645472511829522, −1.14846418186909967931224067904, 2.19891043181301778298131367874, 2.84509937878824010384602672624, 4.65155197149474738079474822359, 5.62051547410178261216929974668, 6.20520920536163684674553283140, 7.10153634117146000614316306224, 7.60289699352125996460944707735, 9.288811842874564832116273688616, 9.734181058138933444048946276068, 10.82751253670234292246231024817

Graph of the ZZ-function along the critical line