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Properties

Label 2-693-231.62-c0-0-2
Degree 22
Conductor 693693
Sign −0.0746−0.997i-0.0746 - 0.997i
Analytic cond. 0.3458520.345852
Root an. cond. 0.5880910.588091
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.44 + 1.04i)2-s + (0.672 + 2.06i)4-s + (−0.951 + 0.309i)7-s + (−0.647 + 1.99i)8-s + (0.156 − 0.987i)11-s + (−1.69 − 0.550i)14-s + (−1.26 + 0.915i)16-s + (1.26 − 1.26i)22-s − 0.907i·23-s + (−0.309 + 0.951i)25-s + (−1.27 − 1.76i)28-s + (−0.610 − 1.87i)29-s − 0.680·32-s + (0.587 + 1.80i)37-s − 1.61i·43-s + (2.14 − 0.340i)44-s + ⋯
L(s)  = 1  + (1.44 + 1.04i)2-s + (0.672 + 2.06i)4-s + (−0.951 + 0.309i)7-s + (−0.647 + 1.99i)8-s + (0.156 − 0.987i)11-s + (−1.69 − 0.550i)14-s + (−1.26 + 0.915i)16-s + (1.26 − 1.26i)22-s − 0.907i·23-s + (−0.309 + 0.951i)25-s + (−1.27 − 1.76i)28-s + (−0.610 − 1.87i)29-s − 0.680·32-s + (0.587 + 1.80i)37-s − 1.61i·43-s + (2.14 − 0.340i)44-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((−0.0746−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0746 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s) L(s)=((−0.0746−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0746 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: −0.0746−0.997i-0.0746 - 0.997i
Analytic conductor: 0.3458520.345852
Root analytic conductor: 0.5880910.588091
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ693(62,⋅)\chi_{693} (62, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 693, ( :0), −0.0746−0.997i)(2,\ 693,\ (\ :0),\ -0.0746 - 0.997i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8058843931.805884393
L(12)L(\frac12) ≈\approx 1.8058843931.805884393
L(1)L(1) 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+(0.951−0.309i)T 1 + (0.951 - 0.309i)T
11 1+(−0.156+0.987i)T 1 + (-0.156 + 0.987i)T
good2 1+(−1.44−1.04i)T+(0.309+0.951i)T2 1 + (-1.44 - 1.04i)T + (0.309 + 0.951i)T^{2}
5 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
13 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
17 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
19 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
23 1+0.907iT−T2 1 + 0.907iT - T^{2}
29 1+(0.610+1.87i)T+(−0.809+0.587i)T2 1 + (0.610 + 1.87i)T + (-0.809 + 0.587i)T^{2}
31 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
37 1+(−0.587−1.80i)T+(−0.809+0.587i)T2 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2}
41 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
43 1+1.61iT−T2 1 + 1.61iT - T^{2}
47 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
53 1+(1.16−1.59i)T+(−0.309−0.951i)T2 1 + (1.16 - 1.59i)T + (-0.309 - 0.951i)T^{2}
59 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
61 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
67 1+1.61T+T2 1 + 1.61T + T^{2}
71 1+(0.183+0.253i)T+(−0.309+0.951i)T2 1 + (0.183 + 0.253i)T + (-0.309 + 0.951i)T^{2}
73 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
79 1+(0.690−0.951i)T+(−0.309−0.951i)T2 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2}
83 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
89 1+T2 1 + T^{2}
97 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)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

−11.25965491571078143744383333678, −9.990883115186801755959936968465, −8.919305908358385731978524818982, −8.016599392683447054595319772026, −7.08552178878075303428054059816, −6.13358018807343265620708184832, −5.80553046026705462399899002223, −4.56478611545197006528119498634, −3.60082808420528854247720005839, −2.74640021440156624103945150881, 1.71686687436873811534454076603, 2.95567458311936212579182386401, 3.83084456724513718354492078213, 4.69571426433541109716661727702, 5.71668115527645180705147482405, 6.59335445368470888233751291219, 7.53069221987970826876998788481, 9.210275745185808957845173796959, 9.884532493081300679133753244821, 10.64299459938258088785807595264

Graph of the ZZ-function along the critical line