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Properties

Label 2-3069-341.278-c0-0-0
Degree 22
Conductor 30693069
Sign −0.794−0.606i-0.794 - 0.606i
Analytic cond. 1.531631.53163
Root an. cond. 1.237591.23759
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  + (−1.30 + 0.951i)2-s + (0.500 − 1.53i)4-s + (−0.809 − 0.587i)5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + 1.61·10-s + (−0.951 − 0.309i)11-s + (−0.587 − 0.809i)13-s + (−0.499 − 1.53i)14-s + (0.951 − 1.30i)17-s + (−1.30 + 0.951i)20-s + (1.53 − 0.499i)22-s + 1.61i·23-s + (1.53 + 0.5i)26-s + (1.30 + 0.951i)28-s + (−0.587 − 0.190i)29-s + ⋯
L(s)  = 1  + (−1.30 + 0.951i)2-s + (0.500 − 1.53i)4-s + (−0.809 − 0.587i)5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + 1.61·10-s + (−0.951 − 0.309i)11-s + (−0.587 − 0.809i)13-s + (−0.499 − 1.53i)14-s + (0.951 − 1.30i)17-s + (−1.30 + 0.951i)20-s + (1.53 − 0.499i)22-s + 1.61i·23-s + (1.53 + 0.5i)26-s + (1.30 + 0.951i)28-s + (−0.587 − 0.190i)29-s + ⋯

Functional equation

Λ(s)=(3069s/2 ΓC(s) L(s)=((−0.794−0.606i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3069s/2 ΓC(s) L(s)=((−0.794−0.606i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30693069    =    32⋅11⋅313^{2} \cdot 11 \cdot 31
Sign: −0.794−0.606i-0.794 - 0.606i
Analytic conductor: 1.531631.53163
Root analytic conductor: 1.237591.23759
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3069(2665,⋅)\chi_{3069} (2665, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3069, ( :0), −0.794−0.606i)(2,\ 3069,\ (\ :0),\ -0.794 - 0.606i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.25355421330.2535542133
L(12)L(\frac12) ≈\approx 0.25355421330.2535542133
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
11 1+(0.951+0.309i)T 1 + (0.951 + 0.309i)T
31 1+(−0.587−0.809i)T 1 + (-0.587 - 0.809i)T
good2 1+(1.30−0.951i)T+(0.309−0.951i)T2 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2}
5 1+(0.809+0.587i)T+(0.309+0.951i)T2 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2}
7 1+(0.309−0.951i)T+(−0.809−0.587i)T2 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2}
13 1+(0.587+0.809i)T+(−0.309+0.951i)T2 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2}
17 1+(−0.951+1.30i)T+(−0.309−0.951i)T2 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2}
19 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
23 1−1.61iT−T2 1 - 1.61iT - T^{2}
29 1+(0.587+0.190i)T+(0.809+0.587i)T2 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2}
37 1+(0.951+0.309i)T+(0.809+0.587i)T2 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2}
41 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
43 1−0.618iT−T2 1 - 0.618iT - T^{2}
47 1+(−0.5−1.53i)T+(−0.809+0.587i)T2 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2}
53 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
59 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
61 1+(−0.587+0.809i)T+(−0.309−0.951i)T2 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2}
67 1+T+T2 1 + T + T^{2}
71 1+(−0.809−0.587i)T+(0.309+0.951i)T2 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2}
73 1+(−0.951−0.309i)T+(0.809+0.587i)T2 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2}
79 1+(0.587+0.809i)T+(−0.309+0.951i)T2 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2}
83 1+(−0.587+0.809i)T+(−0.309−0.951i)T2 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2}
89 1−iT−T2 1 - iT - T^{2}
97 1+(0.809−0.587i)T+(0.309−0.951i)T2 1 + (0.809 - 0.587i)T + (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

−9.119272227960825363721981390616, −8.291197925475405828141452697941, −7.69453705015340843358470588557, −7.45120052596838557540029526650, −6.29150795954412143297019945982, −5.37444358142129514049070959033, −5.13459683596078091549899443059, −3.52390698900113861079359532691, −2.61341267656107250828460673687, −0.993886732549319956964329871475, 0.29867857044336440412353208377, 1.77767250908650464485496770269, 2.71215713454088288102479133771, 3.62299006049498412562529434817, 4.27224228421158183220064988086, 5.55597845420818159164000756557, 6.91881896523529384017115004287, 7.22431981359189494174172147282, 8.106437268297738964955598291871, 8.468056003091619911953639753117

Graph of the ZZ-function along the critical line