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Properties

Label 2-693-7.2-c1-0-11
Degree 22
Conductor 693693
Sign 0.118−0.992i0.118 - 0.992i
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  + (−0.276 + 0.478i)2-s + (0.847 + 1.46i)4-s + (0.795 − 1.37i)5-s + (0.886 + 2.49i)7-s − 2.04·8-s + (0.439 + 0.760i)10-s + (−0.5 − 0.866i)11-s + 2.87·13-s + (−1.43 − 0.264i)14-s + (−1.13 + 1.95i)16-s + (2.41 + 4.18i)17-s + (0.572 − 0.992i)19-s + 2.69·20-s + 0.552·22-s + (−1.82 + 3.16i)23-s + ⋯
L(s)  = 1  + (−0.195 + 0.338i)2-s + (0.423 + 0.733i)4-s + (0.355 − 0.615i)5-s + (0.335 + 0.942i)7-s − 0.721·8-s + (0.138 + 0.240i)10-s + (−0.150 − 0.261i)11-s + 0.798·13-s + (−0.384 − 0.0706i)14-s + (−0.282 + 0.489i)16-s + (0.586 + 1.01i)17-s + (0.131 − 0.227i)19-s + 0.602·20-s + 0.117·22-s + (−0.380 + 0.659i)23-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((0.118−0.992i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+1/2) L(s)=((0.118−0.992i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: 0.118−0.992i0.118 - 0.992i
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), 0.118−0.992i)(2,\ 693,\ (\ :1/2),\ 0.118 - 0.992i)

Particular Values

L(1)L(1) ≈\approx 1.20362+1.06888i1.20362 + 1.06888i
L(12)L(\frac12) ≈\approx 1.20362+1.06888i1.20362 + 1.06888i
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+(−0.886−2.49i)T 1 + (-0.886 - 2.49i)T
11 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good2 1+(0.276−0.478i)T+(−1−1.73i)T2 1 + (0.276 - 0.478i)T + (-1 - 1.73i)T^{2}
5 1+(−0.795+1.37i)T+(−2.5−4.33i)T2 1 + (-0.795 + 1.37i)T + (-2.5 - 4.33i)T^{2}
13 1−2.87T+13T2 1 - 2.87T + 13T^{2}
17 1+(−2.41−4.18i)T+(−8.5+14.7i)T2 1 + (-2.41 - 4.18i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.572+0.992i)T+(−9.5−16.4i)T2 1 + (-0.572 + 0.992i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.82−3.16i)T+(−11.5−19.9i)T2 1 + (1.82 - 3.16i)T + (-11.5 - 19.9i)T^{2}
29 1−0.325T+29T2 1 - 0.325T + 29T^{2}
31 1+(3.22+5.58i)T+(−15.5+26.8i)T2 1 + (3.22 + 5.58i)T + (-15.5 + 26.8i)T^{2}
37 1+(0.847−1.46i)T+(−18.5−32.0i)T2 1 + (0.847 - 1.46i)T + (-18.5 - 32.0i)T^{2}
41 1−4.05T+41T2 1 - 4.05T + 41T^{2}
43 1−4.62T+43T2 1 - 4.62T + 43T^{2}
47 1+(−0.152+0.264i)T+(−23.5−40.7i)T2 1 + (-0.152 + 0.264i)T + (-23.5 - 40.7i)T^{2}
53 1+(−2.85−4.94i)T+(−26.5+45.8i)T2 1 + (-2.85 - 4.94i)T + (-26.5 + 45.8i)T^{2}
59 1+(−5.93−10.2i)T+(−29.5+51.0i)T2 1 + (-5.93 - 10.2i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.886−1.53i)T+(−30.5−52.8i)T2 1 + (0.886 - 1.53i)T + (-30.5 - 52.8i)T^{2}
67 1+(7.63+13.2i)T+(−33.5+58.0i)T2 1 + (7.63 + 13.2i)T + (-33.5 + 58.0i)T^{2}
71 1+9.16T+71T2 1 + 9.16T + 71T^{2}
73 1+(−5.86−10.1i)T+(−36.5+63.2i)T2 1 + (-5.86 - 10.1i)T + (-36.5 + 63.2i)T^{2}
79 1+(−4.35+7.54i)T+(−39.5−68.4i)T2 1 + (-4.35 + 7.54i)T + (-39.5 - 68.4i)T^{2}
83 1+8.40T+83T2 1 + 8.40T + 83T^{2}
89 1+(2.87−4.97i)T+(−44.5−77.0i)T2 1 + (2.87 - 4.97i)T + (-44.5 - 77.0i)T^{2}
97 1+3.65T+97T2 1 + 3.65T + 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

−10.79846967185766954230255160259, −9.484778834714513355720155549284, −8.783988168961956592613725771907, −8.180115812866724977833067417966, −7.33277799332082149020284272925, −5.97717821933176128086367905696, −5.64625144953556547059251116268, −4.13188685754835853126872866666, −2.97729711060326083206686738474, −1.67594575944611683732055197894, 0.971547334627725741580565518022, 2.28225935909294695506267709265, 3.46003492026133032423989807151, 4.81920563112858470432290573431, 5.87468511616766347349932527378, 6.75953848884527165825971240332, 7.46541547881080104872506290985, 8.668928592332088228162735211389, 9.746920777047776826610838528506, 10.35337698482755819994941740305

Graph of the ZZ-function along the critical line