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Properties

Label 2-5733-1.1-c1-0-80
Degree 22
Conductor 57335733
Sign 11
Analytic cond. 45.778245.7782
Root an. cond. 6.765966.76596
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.381·2-s − 1.85·4-s + 2.23·5-s − 1.47·8-s + 0.854·10-s + 3·11-s − 13-s + 3.14·16-s + 7.47·17-s + 3·19-s − 4.14·20-s + 1.14·22-s + 3.76·23-s − 0.381·26-s + 4.47·29-s + 5·31-s + 4.14·32-s + 2.85·34-s − 8.70·37-s + 1.14·38-s − 3.29·40-s − 4.47·41-s − 8·43-s − 5.56·44-s + 1.43·46-s − 1.47·47-s + 1.85·52-s + ⋯
L(s)  = 1  + 0.270·2-s − 0.927·4-s + 0.999·5-s − 0.520·8-s + 0.270·10-s + 0.904·11-s − 0.277·13-s + 0.786·16-s + 1.81·17-s + 0.688·19-s − 0.927·20-s + 0.244·22-s + 0.784·23-s − 0.0749·26-s + 0.830·29-s + 0.898·31-s + 0.732·32-s + 0.489·34-s − 1.43·37-s + 0.185·38-s − 0.520·40-s − 0.698·41-s − 1.21·43-s − 0.838·44-s + 0.211·46-s − 0.214·47-s + 0.257·52-s + ⋯

Functional equation

Λ(s)=(5733s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5733s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57335733    =    32⋅72⋅133^{2} \cdot 7^{2} \cdot 13
Sign: 11
Analytic conductor: 45.778245.7782
Root analytic conductor: 6.765966.76596
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5733, ( :1/2), 1)(2,\ 5733,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5395619072.539561907
L(12)L(\frac12) ≈\approx 2.5395619072.539561907
L(32)L(\frac{3}{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
7 1 1
13 1+T 1 + T
good2 1−0.381T+2T2 1 - 0.381T + 2T^{2}
5 1−2.23T+5T2 1 - 2.23T + 5T^{2}
11 1−3T+11T2 1 - 3T + 11T^{2}
17 1−7.47T+17T2 1 - 7.47T + 17T^{2}
19 1−3T+19T2 1 - 3T + 19T^{2}
23 1−3.76T+23T2 1 - 3.76T + 23T^{2}
29 1−4.47T+29T2 1 - 4.47T + 29T^{2}
31 1−5T+31T2 1 - 5T + 31T^{2}
37 1+8.70T+37T2 1 + 8.70T + 37T^{2}
41 1+4.47T+41T2 1 + 4.47T + 41T^{2}
43 1+8T+43T2 1 + 8T + 43T^{2}
47 1+1.47T+47T2 1 + 1.47T + 47T^{2}
53 1+1.47T+53T2 1 + 1.47T + 53T^{2}
59 1+7.47T+59T2 1 + 7.47T + 59T^{2}
61 1−3T+61T2 1 - 3T + 61T^{2}
67 1+3T+67T2 1 + 3T + 67T^{2}
71 1+8.94T+71T2 1 + 8.94T + 71T^{2}
73 1−10.7T+73T2 1 - 10.7T + 73T^{2}
79 1−10.7T+79T2 1 - 10.7T + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1−2.23T+89T2 1 - 2.23T + 89T^{2}
97 1+17.4T+97T2 1 + 17.4T + 97T^{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

−8.270612109086063232287708959359, −7.39496068555586426443115287178, −6.52839518284144799122193282594, −5.86322008664896671218046157369, −5.17501842036711965255703147720, −4.69777243122186607431825797567, −3.50998178364644846807811630612, −3.11108144127331549652676534880, −1.70417661669501108230058458565, −0.883430952743246814743964384621, 0.883430952743246814743964384621, 1.70417661669501108230058458565, 3.11108144127331549652676534880, 3.50998178364644846807811630612, 4.69777243122186607431825797567, 5.17501842036711965255703147720, 5.86322008664896671218046157369, 6.52839518284144799122193282594, 7.39496068555586426443115287178, 8.270612109086063232287708959359

Graph of the ZZ-function along the critical line