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Properties

Label 2-1575-1.1-c3-0-139
Degree 22
Conductor 15751575
Sign −1-1
Analytic cond. 92.928092.9280
Root an. cond. 9.639919.63991
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 17·4-s − 7·7-s + 45·8-s − 12·11-s − 30·13-s − 35·14-s + 89·16-s − 134·17-s − 92·19-s − 60·22-s + 112·23-s − 150·26-s − 119·28-s + 58·29-s − 224·31-s + 85·32-s − 670·34-s + 146·37-s − 460·38-s − 18·41-s − 340·43-s − 204·44-s + 560·46-s + 208·47-s + 49·49-s − 510·52-s + ⋯
L(s)  = 1  + 1.76·2-s + 17/8·4-s − 0.377·7-s + 1.98·8-s − 0.328·11-s − 0.640·13-s − 0.668·14-s + 1.39·16-s − 1.91·17-s − 1.11·19-s − 0.581·22-s + 1.01·23-s − 1.13·26-s − 0.803·28-s + 0.371·29-s − 1.29·31-s + 0.469·32-s − 3.37·34-s + 0.648·37-s − 1.96·38-s − 0.0685·41-s − 1.20·43-s − 0.698·44-s + 1.79·46-s + 0.645·47-s + 1/7·49-s − 1.36·52-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: yes
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+pT 1 + p T
good2 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
11 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
13 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
17 1+134T+p3T2 1 + 134 T + p^{3} T^{2}
19 1+92T+p3T2 1 + 92 T + p^{3} T^{2}
23 1−112T+p3T2 1 - 112 T + p^{3} T^{2}
29 1−2pT+p3T2 1 - 2 p T + p^{3} T^{2}
31 1+224T+p3T2 1 + 224 T + p^{3} T^{2}
37 1−146T+p3T2 1 - 146 T + p^{3} T^{2}
41 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
43 1+340T+p3T2 1 + 340 T + p^{3} T^{2}
47 1−208T+p3T2 1 - 208 T + p^{3} T^{2}
53 1+754T+p3T2 1 + 754 T + p^{3} T^{2}
59 1+380T+p3T2 1 + 380 T + p^{3} T^{2}
61 1−718T+p3T2 1 - 718 T + p^{3} T^{2}
67 1+412T+p3T2 1 + 412 T + p^{3} T^{2}
71 1−960T+p3T2 1 - 960 T + p^{3} T^{2}
73 1+1066T+p3T2 1 + 1066 T + p^{3} T^{2}
79 1−896T+p3T2 1 - 896 T + p^{3} T^{2}
83 1−436T+p3T2 1 - 436 T + p^{3} T^{2}
89 1−1038T+p3T2 1 - 1038 T + p^{3} T^{2}
97 1−702T+p3T2 1 - 702 T + p^{3} 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

−8.630922165644295098716854860057, −7.44690819838335379183793504096, −6.68391687414205998156653311125, −6.20174460147457692092310364651, −5.07154146795750940103689489110, −4.58773506801867108176623064663, −3.67196439188189314431589556933, −2.68828652888043393862619792301, −1.97790511512788001758041752325, 0, 1.97790511512788001758041752325, 2.68828652888043393862619792301, 3.67196439188189314431589556933, 4.58773506801867108176623064663, 5.07154146795750940103689489110, 6.20174460147457692092310364651, 6.68391687414205998156653311125, 7.44690819838335379183793504096, 8.630922165644295098716854860057

Graph of the ZZ-function along the critical line