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Properties

Label 2-693-77.69-c0-0-1
Degree 22
Conductor 693693
Sign 0.999+0.0237i0.999 + 0.0237i
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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.363i)2-s + (−0.190 + 0.587i)4-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (0.809 + 0.587i)11-s + (−0.190 − 0.587i)14-s + 0.618·22-s − 0.618·23-s + (0.309 + 0.951i)25-s + (0.5 + 0.363i)28-s + (0.5 − 1.53i)29-s − 32-s + (0.190 − 0.587i)37-s − 1.61·43-s + (−0.5 + 0.363i)44-s + ⋯
L(s)  = 1  + (0.5 − 0.363i)2-s + (−0.190 + 0.587i)4-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (0.809 + 0.587i)11-s + (−0.190 − 0.587i)14-s + 0.618·22-s − 0.618·23-s + (0.309 + 0.951i)25-s + (0.5 + 0.363i)28-s + (0.5 − 1.53i)29-s − 32-s + (0.190 − 0.587i)37-s − 1.61·43-s + (−0.5 + 0.363i)44-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((0.999+0.0237i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s) L(s)=((0.999+0.0237i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: 0.999+0.0237i0.999 + 0.0237i
Analytic conductor: 0.3458520.345852
Root analytic conductor: 0.5880910.588091
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ693(685,⋅)\chi_{693} (685, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 693, ( :0), 0.999+0.0237i)(2,\ 693,\ (\ :0),\ 0.999 + 0.0237i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2172690301.217269030
L(12)L(\frac12) ≈\approx 1.2172690301.217269030
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+(−0.309+0.951i)T 1 + (-0.309 + 0.951i)T
11 1+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
good2 1+(−0.5+0.363i)T+(0.309−0.951i)T2 1 + (-0.5 + 0.363i)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+0.618T+T2 1 + 0.618T + T^{2}
29 1+(−0.5+1.53i)T+(−0.809−0.587i)T2 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2}
31 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
37 1+(−0.190+0.587i)T+(−0.809−0.587i)T2 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2}
41 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
43 1+1.61T+T2 1 + 1.61T + T^{2}
47 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
53 1+(1.30−0.951i)T+(0.309−0.951i)T2 1 + (1.30 - 0.951i)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+(1.30+0.951i)T+(0.309+0.951i)T2 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2}
73 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
79 1+(−1.30+0.951i)T+(0.309−0.951i)T2 1 + (-1.30 + 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

−10.83301681733723041346287853852, −9.893517205601291695119718822597, −8.968929824521553842450707112374, −7.925261414363894252373646350795, −7.32474443001207576625097753417, −6.21909408234073838145710325050, −4.81523183367441219737895470928, −4.18889246149609497396793835941, −3.26139941008477391550068681989, −1.78605585050675089750236965703, 1.57042847074569540143355802166, 3.19473303539169765575050987320, 4.46240226692688232180939540174, 5.28591457278548246928516887718, 6.18390267531503340560092388832, 6.79147399468227445010776211233, 8.218766942795135459760244403772, 8.909348253926549395740526823902, 9.779916010640865863200040405013, 10.65062822668939876413655651983

Graph of the ZZ-function along the critical line