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Properties

Label 2-3104-776.579-c0-0-0
Degree 22
Conductor 31043104
Sign 0.645+0.763i0.645 + 0.763i
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 + (−0.357 − 1.05i)17-s + (0.382 − 1.92i)19-s + (0.130 + 0.991i)25-s + (0.582 + 0.241i)27-s + (1.66 − 1.66i)33-s + (−0.0255 − 0.389i)41-s + (1.91 − 0.513i)43-s + (0.793 + 0.608i)49-s + (0.263 + 1.32i)51-s + (−0.767 + 2.26i)57-s + (−0.735 − 1.49i)59-s + (0.128 + 0.0255i)67-s + ⋯
L(s)  = 1  + (−1.20 − 0.158i)3-s + (0.465 + 0.124i)9-s + (−1.17 + 1.53i)11-s + (−0.357 − 1.05i)17-s + (0.382 − 1.92i)19-s + (0.130 + 0.991i)25-s + (0.582 + 0.241i)27-s + (1.66 − 1.66i)33-s + (−0.0255 − 0.389i)41-s + (1.91 − 0.513i)43-s + (0.793 + 0.608i)49-s + (0.263 + 1.32i)51-s + (−0.767 + 2.26i)57-s + (−0.735 − 1.49i)59-s + (0.128 + 0.0255i)67-s + ⋯

Functional equation

Λ(s)=(3104s/2 ΓC(s) L(s)=((0.645+0.763i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3104s/2 ΓC(s) L(s)=((0.645+0.763i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31043104    =    25⋅972^{5} \cdot 97
Sign: 0.645+0.763i0.645 + 0.763i
Analytic conductor: 1.549091.54909
Root analytic conductor: 1.244621.24462
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3104(1743,⋅)\chi_{3104} (1743, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3104, ( :0), 0.645+0.763i)(2,\ 3104,\ (\ :0),\ 0.645 + 0.763i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.61274463740.6127446374
L(12)L(\frac12) ≈\approx 0.61274463740.6127446374
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+(0.357+1.05i)T+(−0.793+0.608i)T2 1 + (0.357 + 1.05i)T + (-0.793 + 0.608i)T^{2}
19 1+(−0.382+1.92i)T+(−0.923−0.382i)T2 1 + (-0.382 + 1.92i)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+(0.0255+0.389i)T+(−0.991+0.130i)T2 1 + (0.0255 + 0.389i)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+(0.735+1.49i)T+(−0.608+0.793i)T2 1 + (0.735 + 1.49i)T + (-0.608 + 0.793i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(−0.128−0.0255i)T+(0.923+0.382i)T2 1 + (-0.128 - 0.0255i)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+(−1.69+0.576i)T+(0.793−0.608i)T2 1 + (-1.69 + 0.576i)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

−9.064113439255463073731553361634, −7.62388050746867452009392159815, −7.26407981494712666190940949133, −6.59993443684376841896732080092, −5.57030714628054103495158076847, −4.95672417875904069999531961061, −4.54992673939177369052001983576, −2.99535726560672902421065261412, −2.14729990935756365874884120293, −0.56227069530545432995740059242, 0.953658272059831046423545549700, 2.46169330247526200483843219064, 3.53684198985613688260036575095, 4.41761473844007297160262335679, 5.44838734703180866555951216326, 5.90953583971881628325245974461, 6.30621032830821716465085268737, 7.59266579118080406475494814212, 8.207734370745482847742486541180, 8.820541639529236783360850651085

Graph of the ZZ-function along the critical line