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Properties

Label 2-3104-776.419-c0-0-0
Degree 22
Conductor 31043104
Sign −0.997+0.0717i-0.997 + 0.0717i
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  + (0.207 − 1.57i)3-s + (−1.46 − 0.392i)9-s + (−1.53 − 1.17i)11-s + (0.172 − 0.349i)17-s + (0.923 − 0.617i)19-s + (−0.991 + 0.130i)25-s + (−0.314 + 0.758i)27-s + (−2.16 + 2.16i)33-s + (−1.09 + 1.25i)41-s + (−0.252 + 0.0675i)43-s + (0.608 − 0.793i)49-s + (−0.514 − 0.343i)51-s + (−0.779 − 1.58i)57-s + (−1.85 − 0.630i)59-s + (0.732 + 1.09i)67-s + ⋯
L(s)  = 1  + (0.207 − 1.57i)3-s + (−1.46 − 0.392i)9-s + (−1.53 − 1.17i)11-s + (0.172 − 0.349i)17-s + (0.923 − 0.617i)19-s + (−0.991 + 0.130i)25-s + (−0.314 + 0.758i)27-s + (−2.16 + 2.16i)33-s + (−1.09 + 1.25i)41-s + (−0.252 + 0.0675i)43-s + (0.608 − 0.793i)49-s + (−0.514 − 0.343i)51-s + (−0.779 − 1.58i)57-s + (−1.85 − 0.630i)59-s + (0.732 + 1.09i)67-s + ⋯

Functional equation

Λ(s)=(3104s/2 ΓC(s) L(s)=((−0.997+0.0717i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0717i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3104s/2 ΓC(s) L(s)=((−0.997+0.0717i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0717i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31043104    =    25⋅972^{5} \cdot 97
Sign: −0.997+0.0717i-0.997 + 0.0717i
Analytic conductor: 1.549091.54909
Root analytic conductor: 1.244621.24462
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3104(1583,⋅)\chi_{3104} (1583, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3104, ( :0), −0.997+0.0717i)(2,\ 3104,\ (\ :0),\ -0.997 + 0.0717i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.91692855130.9169285513
L(12)L(\frac12) ≈\approx 0.91692855130.9169285513
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+(−0.130+0.991i)T 1 + (-0.130 + 0.991i)T
good3 1+(−0.207+1.57i)T+(−0.965−0.258i)T2 1 + (-0.207 + 1.57i)T + (-0.965 - 0.258i)T^{2}
5 1+(0.991−0.130i)T2 1 + (0.991 - 0.130i)T^{2}
7 1+(−0.608+0.793i)T2 1 + (-0.608 + 0.793i)T^{2}
11 1+(1.53+1.17i)T+(0.258+0.965i)T2 1 + (1.53 + 1.17i)T + (0.258 + 0.965i)T^{2}
13 1+(−0.991+0.130i)T2 1 + (-0.991 + 0.130i)T^{2}
17 1+(−0.172+0.349i)T+(−0.608−0.793i)T2 1 + (-0.172 + 0.349i)T + (-0.608 - 0.793i)T^{2}
19 1+(−0.923+0.617i)T+(0.382−0.923i)T2 1 + (-0.923 + 0.617i)T + (0.382 - 0.923i)T^{2}
23 1+(−0.793+0.608i)T2 1 + (-0.793 + 0.608i)T^{2}
29 1+(−0.130−0.991i)T2 1 + (-0.130 - 0.991i)T^{2}
31 1+(−0.965−0.258i)T2 1 + (-0.965 - 0.258i)T^{2}
37 1+(0.793+0.608i)T2 1 + (0.793 + 0.608i)T^{2}
41 1+(1.09−1.25i)T+(−0.130−0.991i)T2 1 + (1.09 - 1.25i)T + (-0.130 - 0.991i)T^{2}
43 1+(0.252−0.0675i)T+(0.866−0.5i)T2 1 + (0.252 - 0.0675i)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+(1.85+0.630i)T+(0.793+0.608i)T2 1 + (1.85 + 0.630i)T + (0.793 + 0.608i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(−0.732−1.09i)T+(−0.382+0.923i)T2 1 + (-0.732 - 1.09i)T + (-0.382 + 0.923i)T^{2}
71 1+(0.130−0.991i)T2 1 + (0.130 - 0.991i)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+(0.576+0.284i)T+(0.608+0.793i)T2 1 + (0.576 + 0.284i)T + (0.608 + 0.793i)T^{2}
89 1+(−1.83+0.758i)T+(0.707−0.707i)T2 1 + (-1.83 + 0.758i)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

−8.201171545081573705160975744551, −7.79200902214044221039846445292, −7.17925187341987224003327308683, −6.30669265852939503182811817787, −5.63981515693896875643533856751, −4.89460561593206780591663291239, −3.31079939952567523199389457949, −2.77031535478850142283710161826, −1.74475606756843662744994391208, −0.51618542830514667659849222074, 1.99713116621883188264679205476, 3.01788319148012743773732324003, 3.81395619616716789500916914965, 4.62198138137240264340113371652, 5.25140682587782597646847139384, 5.87060502183431735992020200982, 7.21213442690608933470124567239, 7.84811585891493344156259040458, 8.574053671124547759848465132032, 9.552830848650944757394277582095

Graph of the ZZ-function along the critical line