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Properties

Label 2-3104-776.259-c0-0-0
Degree 22
Conductor 31043104
Sign −0.523−0.851i-0.523 - 0.851i
Analytic cond. 1.549091.54909
Root an. cond. 1.244621.24462
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  + (−1.20 + 0.158i)3-s + (0.465 − 0.124i)9-s + (1.17 + 1.53i)11-s + (−1.57 − 0.534i)17-s + (−0.382 + 0.0761i)19-s + (−0.130 + 0.991i)25-s + (0.582 − 0.241i)27-s + (−1.66 − 1.66i)33-s + (1.95 + 0.128i)41-s + (−1.91 − 0.513i)43-s + (−0.793 + 0.608i)49-s + (1.98 + 0.394i)51-s + (0.449 − 0.152i)57-s + (−0.996 − 0.491i)59-s + (0.389 + 1.95i)67-s + ⋯
L(s)  = 1  + (−1.20 + 0.158i)3-s + (0.465 − 0.124i)9-s + (1.17 + 1.53i)11-s + (−1.57 − 0.534i)17-s + (−0.382 + 0.0761i)19-s + (−0.130 + 0.991i)25-s + (0.582 − 0.241i)27-s + (−1.66 − 1.66i)33-s + (1.95 + 0.128i)41-s + (−1.91 − 0.513i)43-s + (−0.793 + 0.608i)49-s + (1.98 + 0.394i)51-s + (0.449 − 0.152i)57-s + (−0.996 − 0.491i)59-s + (0.389 + 1.95i)67-s + ⋯

Functional equation

Λ(s)=(3104s/2 ΓC(s) L(s)=((−0.523−0.851i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3104s/2 ΓC(s) L(s)=((−0.523−0.851i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31043104    =    25⋅972^{5} \cdot 97
Sign: −0.523−0.851i-0.523 - 0.851i
Analytic conductor: 1.549091.54909
Root analytic conductor: 1.244621.24462
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3104(1423,⋅)\chi_{3104} (1423, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3104, ( :0), −0.523−0.851i)(2,\ 3104,\ (\ :0),\ -0.523 - 0.851i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.52003572660.5200357266
L(12)L(\frac12) ≈\approx 0.52003572660.5200357266
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)
bad2 1 1
97 1+(0.991−0.130i)T 1 + (0.991 - 0.130i)T
good3 1+(1.20−0.158i)T+(0.965−0.258i)T2 1 + (1.20 - 0.158i)T + (0.965 - 0.258i)T^{2}
5 1+(0.130−0.991i)T2 1 + (0.130 - 0.991i)T^{2}
7 1+(0.793−0.608i)T2 1 + (0.793 - 0.608i)T^{2}
11 1+(−1.17−1.53i)T+(−0.258+0.965i)T2 1 + (-1.17 - 1.53i)T + (-0.258 + 0.965i)T^{2}
13 1+(−0.130+0.991i)T2 1 + (-0.130 + 0.991i)T^{2}
17 1+(1.57+0.534i)T+(0.793+0.608i)T2 1 + (1.57 + 0.534i)T + (0.793 + 0.608i)T^{2}
19 1+(0.382−0.0761i)T+(0.923−0.382i)T2 1 + (0.382 - 0.0761i)T + (0.923 - 0.382i)T^{2}
23 1+(−0.608+0.793i)T2 1 + (-0.608 + 0.793i)T^{2}
29 1+(0.991+0.130i)T2 1 + (0.991 + 0.130i)T^{2}
31 1+(0.965−0.258i)T2 1 + (0.965 - 0.258i)T^{2}
37 1+(0.608+0.793i)T2 1 + (0.608 + 0.793i)T^{2}
41 1+(−1.95−0.128i)T+(0.991+0.130i)T2 1 + (-1.95 - 0.128i)T + (0.991 + 0.130i)T^{2}
43 1+(1.91+0.513i)T+(0.866+0.5i)T2 1 + (1.91 + 0.513i)T + (0.866 + 0.5i)T^{2}
47 1+iT2 1 + iT^{2}
53 1+(0.258+0.965i)T2 1 + (0.258 + 0.965i)T^{2}
59 1+(0.996+0.491i)T+(0.608+0.793i)T2 1 + (0.996 + 0.491i)T + (0.608 + 0.793i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(−0.389−1.95i)T+(−0.923+0.382i)T2 1 + (-0.389 - 1.95i)T + (-0.923 + 0.382i)T^{2}
71 1+(−0.991+0.130i)T2 1 + (-0.991 + 0.130i)T^{2}
73 1+(0.366−1.36i)T+(−0.866−0.5i)T2 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2}
79 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
83 1+(0.284−0.837i)T+(−0.793−0.608i)T2 1 + (0.284 - 0.837i)T + (-0.793 - 0.608i)T^{2}
89 1+(0.0999−0.241i)T+(−0.707−0.707i)T2 1 + (0.0999 - 0.241i)T + (-0.707 - 0.707i)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

−9.307464555120229540639932222822, −8.488671042010364369580826397183, −7.31217236455379207208531378313, −6.77705909990225858950282082965, −6.21618131754265106212567244006, −5.23439682333590426331139296009, −4.56539998902024891229202916127, −3.97742856967178051401590773994, −2.49571177421627777904925971508, −1.42022198341106382219542678069, 0.38693582084575045672180767396, 1.67775310415337207426577502824, 3.03722089307497564143859685834, 4.08076183343830602073085637365, 4.78519407278151713021963601608, 5.83631533123921610778230811832, 6.40462594845237490423644904559, 6.62637121368448101958307015006, 7.943402427858433962542001439686, 8.701805397837258835727263812305

Graph of the ZZ-function along the critical line