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Properties

Label 2-3104-776.387-c0-0-4
Degree 22
Conductor 31043104
Sign 11
Analytic cond. 1.549091.54909
Root an. cond. 1.244621.24462
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·3-s + 1.90·5-s + 1.17·7-s − 0.618·9-s − 1.61·11-s − 1.17·15-s − 0.726·21-s + 1.90·23-s + 2.61·25-s + 27-s − 1.17·29-s + 1.00·33-s + 2.23·35-s + 1.17·37-s + 0.618·43-s − 1.17·45-s + 0.381·49-s − 3.07·55-s − 0.726·63-s − 1.17·69-s + 0.618·73-s − 1.61·75-s − 1.90·77-s + 0.726·87-s − 1.61·89-s − 97-s + 0.999·99-s + ⋯
L(s)  = 1  − 0.618·3-s + 1.90·5-s + 1.17·7-s − 0.618·9-s − 1.61·11-s − 1.17·15-s − 0.726·21-s + 1.90·23-s + 2.61·25-s + 27-s − 1.17·29-s + 1.00·33-s + 2.23·35-s + 1.17·37-s + 0.618·43-s − 1.17·45-s + 0.381·49-s − 3.07·55-s − 0.726·63-s − 1.17·69-s + 0.618·73-s − 1.61·75-s − 1.90·77-s + 0.726·87-s − 1.61·89-s − 97-s + 0.999·99-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 31043104    =    25⋅972^{5} \cdot 97
Sign: 11
Analytic conductor: 1.549091.54909
Root analytic conductor: 1.244621.24462
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3104(1551,⋅)\chi_{3104} (1551, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3104, ( :0), 1)(2,\ 3104,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4713510631.471351063
L(12)L(\frac12) ≈\approx 1.4713510631.471351063
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+T 1 + T
good3 1+0.618T+T2 1 + 0.618T + T^{2}
5 1−1.90T+T2 1 - 1.90T + T^{2}
7 1−1.17T+T2 1 - 1.17T + T^{2}
11 1+1.61T+T2 1 + 1.61T + T^{2}
13 1+T2 1 + T^{2}
17 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1−1.90T+T2 1 - 1.90T + T^{2}
29 1+1.17T+T2 1 + 1.17T + T^{2}
31 1−T2 1 - T^{2}
37 1−1.17T+T2 1 - 1.17T + T^{2}
41 1−T2 1 - T^{2}
43 1−0.618T+T2 1 - 0.618T + T^{2}
47 1−T2 1 - T^{2}
53 1−T2 1 - T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1−T2 1 - T^{2}
71 1+T2 1 + T^{2}
73 1−0.618T+T2 1 - 0.618T + T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1+1.61T+T2 1 + 1.61T + 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

−8.954703446201520601902076244677, −8.188248028110388130480359521949, −7.34182668652713979329700221657, −6.41844317506333171614337114532, −5.50686430980829053919468564173, −5.36757934358516239986677384224, −4.67998348222682603225291530528, −2.88703602176581564913693923500, −2.31311234255746116173229814516, −1.21987291166824467592526166629, 1.21987291166824467592526166629, 2.31311234255746116173229814516, 2.88703602176581564913693923500, 4.67998348222682603225291530528, 5.36757934358516239986677384224, 5.50686430980829053919468564173, 6.41844317506333171614337114532, 7.34182668652713979329700221657, 8.188248028110388130480359521949, 8.954703446201520601902076244677

Graph of the ZZ-function along the critical line