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Properties

Label 2-1575-1.1-c3-0-109
Degree 22
Conductor 15751575
Sign −1-1
Analytic cond. 92.928092.9280
Root an. cond. 9.639919.63991
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.19·2-s − 6.56·4-s − 7·7-s − 17.4·8-s − 4.52·11-s + 14.5·13-s − 8.37·14-s + 31.7·16-s − 15.5·17-s − 14.8·19-s − 5.41·22-s + 166.·23-s + 17.4·26-s + 45.9·28-s + 194.·29-s + 104.·31-s + 177.·32-s − 18.6·34-s − 47.5·37-s − 17.7·38-s − 378.·41-s + 194.·43-s + 29.7·44-s + 199.·46-s − 488.·47-s + 49·49-s − 95.7·52-s + ⋯
L(s)  = 1  + 0.422·2-s − 0.821·4-s − 0.377·7-s − 0.770·8-s − 0.124·11-s + 0.310·13-s − 0.159·14-s + 0.495·16-s − 0.221·17-s − 0.179·19-s − 0.0524·22-s + 1.51·23-s + 0.131·26-s + 0.310·28-s + 1.24·29-s + 0.605·31-s + 0.979·32-s − 0.0938·34-s − 0.211·37-s − 0.0758·38-s − 1.44·41-s + 0.691·43-s + 0.101·44-s + 0.639·46-s − 1.51·47-s + 0.142·49-s − 0.255·52-s + ⋯

Functional equation

Λ(s)=(1575s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1575s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15751575    =    32⋅52⋅73^{2} \cdot 5^{2} \cdot 7
Sign: −1-1
Analytic conductor: 92.928092.9280
Root analytic conductor: 9.639919.63991
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1575, ( :3/2), −1)(2,\ 1575,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) 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)
bad3 1 1
5 1 1
7 1+7T 1 + 7T
good2 1−1.19T+8T2 1 - 1.19T + 8T^{2}
11 1+4.52T+1.33e3T2 1 + 4.52T + 1.33e3T^{2}
13 1−14.5T+2.19e3T2 1 - 14.5T + 2.19e3T^{2}
17 1+15.5T+4.91e3T2 1 + 15.5T + 4.91e3T^{2}
19 1+14.8T+6.85e3T2 1 + 14.8T + 6.85e3T^{2}
23 1−166.T+1.21e4T2 1 - 166.T + 1.21e4T^{2}
29 1−194.T+2.43e4T2 1 - 194.T + 2.43e4T^{2}
31 1−104.T+2.97e4T2 1 - 104.T + 2.97e4T^{2}
37 1+47.5T+5.06e4T2 1 + 47.5T + 5.06e4T^{2}
41 1+378.T+6.89e4T2 1 + 378.T + 6.89e4T^{2}
43 1−194.T+7.95e4T2 1 - 194.T + 7.95e4T^{2}
47 1+488.T+1.03e5T2 1 + 488.T + 1.03e5T^{2}
53 1+316.T+1.48e5T2 1 + 316.T + 1.48e5T^{2}
59 1+350.T+2.05e5T2 1 + 350.T + 2.05e5T^{2}
61 1−115.T+2.26e5T2 1 - 115.T + 2.26e5T^{2}
67 1+434.T+3.00e5T2 1 + 434.T + 3.00e5T^{2}
71 1−410.T+3.57e5T2 1 - 410.T + 3.57e5T^{2}
73 1−464.T+3.89e5T2 1 - 464.T + 3.89e5T^{2}
79 1+290.T+4.93e5T2 1 + 290.T + 4.93e5T^{2}
83 1+578.T+5.71e5T2 1 + 578.T + 5.71e5T^{2}
89 1−937.T+7.04e5T2 1 - 937.T + 7.04e5T^{2}
97 1+839.T+9.12e5T2 1 + 839.T + 9.12e5T^{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

−8.687496907191919963543619272677, −8.054031040373261777762029274148, −6.84737695098697817801058802849, −6.20642830598497974048347734189, −5.14334382571885853409104576655, −4.59197763588451983805407658445, −3.52074950236082126130273708813, −2.79404003475741885108642862281, −1.18601180477300205568132783563, 0, 1.18601180477300205568132783563, 2.79404003475741885108642862281, 3.52074950236082126130273708813, 4.59197763588451983805407658445, 5.14334382571885853409104576655, 6.20642830598497974048347734189, 6.84737695098697817801058802849, 8.054031040373261777762029274148, 8.687496907191919963543619272677

Graph of the ZZ-function along the critical line