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Properties

Label 2-3104-776.99-c0-0-0
Degree 22
Conductor 31043104
Sign 0.308−0.951i0.308 - 0.951i
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 − 0.158i)3-s + (−0.241 + 0.900i)9-s + (−0.0675 − 0.513i)11-s + (−0.128 + 1.95i)17-s + (−0.923 + 1.38i)19-s + (−0.608 + 0.793i)25-s + (0.192 + 0.465i)27-s + (−0.0955 − 0.0955i)33-s + (1.05 + 0.357i)41-s + (−0.410 − 1.53i)43-s + (0.991 − 0.130i)49-s + (0.284 + 0.425i)51-s + (0.0283 + 0.433i)57-s + (0.257 − 0.293i)59-s + (1.57 + 1.05i)67-s + ⋯
L(s)  = 1  + (0.207 − 0.158i)3-s + (−0.241 + 0.900i)9-s + (−0.0675 − 0.513i)11-s + (−0.128 + 1.95i)17-s + (−0.923 + 1.38i)19-s + (−0.608 + 0.793i)25-s + (0.192 + 0.465i)27-s + (−0.0955 − 0.0955i)33-s + (1.05 + 0.357i)41-s + (−0.410 − 1.53i)43-s + (0.991 − 0.130i)49-s + (0.284 + 0.425i)51-s + (0.0283 + 0.433i)57-s + (0.257 − 0.293i)59-s + (1.57 + 1.05i)67-s + ⋯

Functional equation

Λ(s)=(3104s/2 ΓC(s) L(s)=((0.308−0.951i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3104s/2 ΓC(s) L(s)=((0.308−0.951i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31043104    =    25⋅972^{5} \cdot 97
Sign: 0.308−0.951i0.308 - 0.951i
Analytic conductor: 1.549091.54909
Root analytic conductor: 1.244621.24462
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3104(1263,⋅)\chi_{3104} (1263, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3104, ( :0), 0.308−0.951i)(2,\ 3104,\ (\ :0),\ 0.308 - 0.951i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1005959831.100595983
L(12)L(\frac12) ≈\approx 1.1005959831.100595983
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+(0.793−0.608i)T 1 + (0.793 - 0.608i)T
good3 1+(−0.207+0.158i)T+(0.258−0.965i)T2 1 + (-0.207 + 0.158i)T + (0.258 - 0.965i)T^{2}
5 1+(0.608−0.793i)T2 1 + (0.608 - 0.793i)T^{2}
7 1+(−0.991+0.130i)T2 1 + (-0.991 + 0.130i)T^{2}
11 1+(0.0675+0.513i)T+(−0.965+0.258i)T2 1 + (0.0675 + 0.513i)T + (-0.965 + 0.258i)T^{2}
13 1+(−0.608+0.793i)T2 1 + (-0.608 + 0.793i)T^{2}
17 1+(0.128−1.95i)T+(−0.991−0.130i)T2 1 + (0.128 - 1.95i)T + (-0.991 - 0.130i)T^{2}
19 1+(0.923−1.38i)T+(−0.382−0.923i)T2 1 + (0.923 - 1.38i)T + (-0.382 - 0.923i)T^{2}
23 1+(0.130−0.991i)T2 1 + (0.130 - 0.991i)T^{2}
29 1+(0.793+0.608i)T2 1 + (0.793 + 0.608i)T^{2}
31 1+(0.258−0.965i)T2 1 + (0.258 - 0.965i)T^{2}
37 1+(−0.130−0.991i)T2 1 + (-0.130 - 0.991i)T^{2}
41 1+(−1.05−0.357i)T+(0.793+0.608i)T2 1 + (-1.05 - 0.357i)T + (0.793 + 0.608i)T^{2}
43 1+(0.410+1.53i)T+(−0.866+0.5i)T2 1 + (0.410 + 1.53i)T + (-0.866 + 0.5i)T^{2}
47 1+iT2 1 + iT^{2}
53 1+(0.965+0.258i)T2 1 + (0.965 + 0.258i)T^{2}
59 1+(−0.257+0.293i)T+(−0.130−0.991i)T2 1 + (-0.257 + 0.293i)T + (-0.130 - 0.991i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(−1.57−1.05i)T+(0.382+0.923i)T2 1 + (-1.57 - 1.05i)T + (0.382 + 0.923i)T^{2}
71 1+(−0.793+0.608i)T2 1 + (-0.793 + 0.608i)T^{2}
73 1+(−1.36+0.366i)T+(0.866−0.5i)T2 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2}
79 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
83 1+(1.50+0.0983i)T+(0.991+0.130i)T2 1 + (1.50 + 0.0983i)T + (0.991 + 0.130i)T^{2}
89 1+(1.12+0.465i)T+(0.707+0.707i)T2 1 + (1.12 + 0.465i)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

−8.736347362350989212313062800588, −8.284818384137713390637196575219, −7.75966656881997772296137487313, −6.76737189081060869401591275106, −5.86974362211968950700419100202, −5.44501787359434701888215029539, −4.14588438121690093189640279078, −3.62754927260318928381352669345, −2.34052816441802697697617696023, −1.59984221475089465836789092206, 0.64576724806231953065861426128, 2.34925109558118133221651070436, 2.94914985468286081625294431608, 4.14369019923536391000171761255, 4.72232347847113313240748201334, 5.67971229344761344918188463177, 6.65715725576688099690521645807, 7.07588681225569484351274685595, 8.051014113988557240143178145862, 8.841132662396466021991819439331

Graph of the ZZ-function along the critical line