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Properties

Label 2-693-11.3-c1-0-14
Degree 22
Conductor 693693
Sign 0.828−0.560i0.828 - 0.560i
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  + (−2.16 + 1.57i)2-s + (1.60 − 4.93i)4-s + (0.946 + 0.687i)5-s + (0.309 − 0.951i)7-s + (2.64 + 8.12i)8-s − 3.13·10-s + (3.31 + 0.0332i)11-s + (1.36 − 0.989i)13-s + (0.828 + 2.54i)14-s + (−10.1 − 7.36i)16-s + (−4.52 − 3.28i)17-s + (1.34 + 4.14i)19-s + (4.91 − 3.56i)20-s + (−7.24 + 5.15i)22-s + 0.119·23-s + ⋯
L(s)  = 1  + (−1.53 + 1.11i)2-s + (0.801 − 2.46i)4-s + (0.423 + 0.307i)5-s + (0.116 − 0.359i)7-s + (0.933 + 2.87i)8-s − 0.992·10-s + (0.999 + 0.0100i)11-s + (0.377 − 0.274i)13-s + (0.221 + 0.681i)14-s + (−2.53 − 1.84i)16-s + (−1.09 − 0.797i)17-s + (0.308 + 0.950i)19-s + (1.09 − 0.797i)20-s + (−1.54 + 1.09i)22-s + 0.0249·23-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((0.828−0.560i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+1/2) L(s)=((0.828−0.560i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: 0.828−0.560i0.828 - 0.560i
Analytic conductor: 5.533635.53363
Root analytic conductor: 2.352362.35236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ693(190,⋅)\chi_{693} (190, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 693, ( :1/2), 0.828−0.560i)(2,\ 693,\ (\ :1/2),\ 0.828 - 0.560i)

Particular Values

L(1)L(1) ≈\approx 0.766090+0.234680i0.766090 + 0.234680i
L(12)L(\frac12) ≈\approx 0.766090+0.234680i0.766090 + 0.234680i
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+(−3.31−0.0332i)T 1 + (-3.31 - 0.0332i)T
good2 1+(2.16−1.57i)T+(0.618−1.90i)T2 1 + (2.16 - 1.57i)T + (0.618 - 1.90i)T^{2}
5 1+(−0.946−0.687i)T+(1.54+4.75i)T2 1 + (-0.946 - 0.687i)T + (1.54 + 4.75i)T^{2}
13 1+(−1.36+0.989i)T+(4.01−12.3i)T2 1 + (-1.36 + 0.989i)T + (4.01 - 12.3i)T^{2}
17 1+(4.52+3.28i)T+(5.25+16.1i)T2 1 + (4.52 + 3.28i)T + (5.25 + 16.1i)T^{2}
19 1+(−1.34−4.14i)T+(−15.3+11.1i)T2 1 + (-1.34 - 4.14i)T + (-15.3 + 11.1i)T^{2}
23 1−0.119T+23T2 1 - 0.119T + 23T^{2}
29 1+(−1.35+4.18i)T+(−23.4−17.0i)T2 1 + (-1.35 + 4.18i)T + (-23.4 - 17.0i)T^{2}
31 1+(−5.10+3.71i)T+(9.57−29.4i)T2 1 + (-5.10 + 3.71i)T + (9.57 - 29.4i)T^{2}
37 1+(−1.80+5.56i)T+(−29.9−21.7i)T2 1 + (-1.80 + 5.56i)T + (-29.9 - 21.7i)T^{2}
41 1+(−3.40−10.4i)T+(−33.1+24.0i)T2 1 + (-3.40 - 10.4i)T + (-33.1 + 24.0i)T^{2}
43 1−5.97T+43T2 1 - 5.97T + 43T^{2}
47 1+(3.88+11.9i)T+(−38.0+27.6i)T2 1 + (3.88 + 11.9i)T + (-38.0 + 27.6i)T^{2}
53 1+(−0.337+0.245i)T+(16.3−50.4i)T2 1 + (-0.337 + 0.245i)T + (16.3 - 50.4i)T^{2}
59 1+(−0.223+0.688i)T+(−47.7−34.6i)T2 1 + (-0.223 + 0.688i)T + (-47.7 - 34.6i)T^{2}
61 1+(−7.16−5.20i)T+(18.8+58.0i)T2 1 + (-7.16 - 5.20i)T + (18.8 + 58.0i)T^{2}
67 1−8.78T+67T2 1 - 8.78T + 67T^{2}
71 1+(−2.59−1.88i)T+(21.9+67.5i)T2 1 + (-2.59 - 1.88i)T + (21.9 + 67.5i)T^{2}
73 1+(4.50−13.8i)T+(−59.0−42.9i)T2 1 + (4.50 - 13.8i)T + (-59.0 - 42.9i)T^{2}
79 1+(11.7−8.50i)T+(24.4−75.1i)T2 1 + (11.7 - 8.50i)T + (24.4 - 75.1i)T^{2}
83 1+(2.79+2.03i)T+(25.6+78.9i)T2 1 + (2.79 + 2.03i)T + (25.6 + 78.9i)T^{2}
89 1−12.0T+89T2 1 - 12.0T + 89T^{2}
97 1+(−4.85+3.52i)T+(29.9−92.2i)T2 1 + (-4.85 + 3.52i)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

−10.01556123185851806720302416348, −9.718669392792579032140081459985, −8.707417579188163167979818101866, −8.031971517775735823446822610128, −7.06926807278559182820884950947, −6.40103396870981102275331658760, −5.69609889909600503504544883253, −4.28669975876020865066540446818, −2.26417097098117036783469837799, −0.870390150104786361044980200208, 1.16091146925074487161667614600, 2.14938157239672827816254389845, 3.36627286710029670828603764490, 4.55334925614743502172767898110, 6.26568889539609276646858333811, 7.11805223836924605309123476807, 8.260477239793071898488347268690, 9.092539660443977554399264856631, 9.219979525033223109108552599301, 10.37281354811190948018766427285

Graph of the ZZ-function along the critical line