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Properties

Label 2-3104-776.35-c0-0-0
Degree 22
Conductor 31043104
Sign −0.421−0.907i-0.421 - 0.907i
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  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)11-s + (−1 + 1.73i)17-s − 2·19-s + (−0.5 + 0.866i)25-s − 27-s + 0.999·33-s + (−1 − 1.73i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)49-s + 1.99·51-s + (1 + 1.73i)57-s + (1 + 1.73i)59-s + 67-s + (−1 − 1.73i)73-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)11-s + (−1 + 1.73i)17-s − 2·19-s + (−0.5 + 0.866i)25-s − 27-s + 0.999·33-s + (−1 − 1.73i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)49-s + 1.99·51-s + (1 + 1.73i)57-s + (1 + 1.73i)59-s + 67-s + (−1 − 1.73i)73-s + ⋯

Functional equation

Λ(s)=(3104s/2 ΓC(s) L(s)=((−0.421−0.907i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3104s/2 ΓC(s) L(s)=((−0.421−0.907i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31043104    =    25⋅972^{5} \cdot 97
Sign: −0.421−0.907i-0.421 - 0.907i
Analytic conductor: 1.549091.54909
Root analytic conductor: 1.244621.24462
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3104(1199,⋅)\chi_{3104} (1199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3104, ( :0), −0.421−0.907i)(2,\ 3104,\ (\ :0),\ -0.421 - 0.907i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.24503950400.2450395040
L(12)L(\frac12) ≈\approx 0.24503950400.2450395040
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good3 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
5 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
7 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
13 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
17 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
19 1+2T+T2 1 + 2T + T^{2}
23 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
41 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
43 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1−T2 1 - T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1−T+T2 1 - T + T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
79 1−T2 1 - T^{2}
83 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
89 1+T+T2 1 + T + 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

−8.879787274606681813387442039881, −8.392242314109835285515641949561, −7.44059513242000742027632490418, −6.84034837741633518865792693516, −6.22481607868057424420343733451, −5.51445091251453794884501302710, −4.39156294253239848070930361960, −3.77528068949820579157003501416, −2.18803829276748699923283381460, −1.70824942684339872176437601431, 0.14588487962323042740174787256, 2.09794552152422887479601829294, 3.02689993280511953731454274230, 4.18535654232123456682614174734, 4.72011249489251468241012098800, 5.42957215311863395141389801652, 6.36238595225842379503116449729, 6.94693610498858726653946230003, 8.211157577544965359464925685891, 8.480831576671176618475018414355

Graph of the ZZ-function along the critical line