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Properties

Label 2-693-77.20-c0-0-1
Degree 22
Conductor 693693
Sign 0.352+0.935i0.352 + 0.935i
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  + (0.363 − 1.11i)2-s + (−0.309 − 0.224i)4-s + (0.809 + 0.587i)7-s + (0.587 − 0.427i)8-s + (−0.951 + 0.309i)11-s + (0.951 − 0.690i)14-s + (−0.381 − 1.17i)16-s + 1.17i·22-s + 1.17·23-s + (−0.809 + 0.587i)25-s + (−0.118 − 0.363i)28-s + (−1.53 − 1.11i)29-s − 0.726·32-s + (−1.30 − 0.951i)37-s − 0.618·43-s + (0.363 + 0.118i)44-s + ⋯
L(s)  = 1  + (0.363 − 1.11i)2-s + (−0.309 − 0.224i)4-s + (0.809 + 0.587i)7-s + (0.587 − 0.427i)8-s + (−0.951 + 0.309i)11-s + (0.951 − 0.690i)14-s + (−0.381 − 1.17i)16-s + 1.17i·22-s + 1.17·23-s + (−0.809 + 0.587i)25-s + (−0.118 − 0.363i)28-s + (−1.53 − 1.11i)29-s − 0.726·32-s + (−1.30 − 0.951i)37-s − 0.618·43-s + (0.363 + 0.118i)44-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((0.352+0.935i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s) L(s)=((0.352+0.935i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: 0.352+0.935i0.352 + 0.935i
Analytic conductor: 0.3458520.345852
Root analytic conductor: 0.5880910.588091
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ693(559,⋅)\chi_{693} (559, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 693, ( :0), 0.352+0.935i)(2,\ 693,\ (\ :0),\ 0.352 + 0.935i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2556150661.255615066
L(12)L(\frac12) ≈\approx 1.2556150661.255615066
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+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
11 1+(0.951−0.309i)T 1 + (0.951 - 0.309i)T
good2 1+(−0.363+1.11i)T+(−0.809−0.587i)T2 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2}
5 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
13 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
17 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
19 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
23 1−1.17T+T2 1 - 1.17T + T^{2}
29 1+(1.53+1.11i)T+(0.309+0.951i)T2 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2}
31 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
37 1+(1.30+0.951i)T+(0.309+0.951i)T2 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2}
41 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
43 1+0.618T+T2 1 + 0.618T + T^{2}
47 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
53 1+(0.587−1.80i)T+(−0.809−0.587i)T2 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2}
59 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
61 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
67 1−0.618T+T2 1 - 0.618T + T^{2}
71 1+(−0.587−1.80i)T+(−0.809+0.587i)T2 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2}
73 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
79 1+(−0.190+0.587i)T+(−0.809−0.587i)T2 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2}
83 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)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

−10.82504240809508026269838967272, −9.864678312453935741675008658535, −9.004621545635098128725605270342, −7.84967796070815818449237866688, −7.21413152871636226217493591879, −5.66116108082301288702986247849, −4.89345401454879941305756148968, −3.80749282235803645485132562817, −2.63228007694156265733524050160, −1.74574801764768563076769334235, 1.85375757083910014426667130809, 3.55422686682421693938063198499, 4.91986103956663797703720217533, 5.29141637295304500988309655485, 6.50190373705563254892509928295, 7.31741051720440190884784049579, 7.983519422038585150805871565737, 8.732277150523890300205766232812, 10.09061763107224981147021016718, 10.88493681460247936380096952156

Graph of the ZZ-function along the critical line