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Properties

Label 2-5733-1.1-c1-0-110
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  + 2.61·2-s + 4.85·4-s − 2.23·5-s + 7.47·8-s − 5.85·10-s + 3·11-s − 13-s + 9.85·16-s − 1.47·17-s + 3·19-s − 10.8·20-s + 7.85·22-s + 8.23·23-s − 2.61·26-s − 4.47·29-s + 5·31-s + 10.8·32-s − 3.85·34-s + 4.70·37-s + 7.85·38-s − 16.7·40-s + 4.47·41-s − 8·43-s + 14.5·44-s + 21.5·46-s + 7.47·47-s − 4.85·52-s + ⋯
L(s)  = 1  + 1.85·2-s + 2.42·4-s − 0.999·5-s + 2.64·8-s − 1.85·10-s + 0.904·11-s − 0.277·13-s + 2.46·16-s − 0.357·17-s + 0.688·19-s − 2.42·20-s + 1.67·22-s + 1.71·23-s − 0.513·26-s − 0.830·29-s + 0.898·31-s + 1.91·32-s − 0.660·34-s + 0.774·37-s + 1.27·38-s − 2.64·40-s + 0.698·41-s − 1.21·43-s + 2.19·44-s + 3.17·46-s + 1.08·47-s − 0.673·52-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 6.2405072426.240507242
L(12)L(\frac12) ≈\approx 6.2405072426.240507242
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 1−2.61T+2T2 1 - 2.61T + 2T^{2}
5 1+2.23T+5T2 1 + 2.23T + 5T^{2}
11 1−3T+11T2 1 - 3T + 11T^{2}
17 1+1.47T+17T2 1 + 1.47T + 17T^{2}
19 1−3T+19T2 1 - 3T + 19T^{2}
23 1−8.23T+23T2 1 - 8.23T + 23T^{2}
29 1+4.47T+29T2 1 + 4.47T + 29T^{2}
31 1−5T+31T2 1 - 5T + 31T^{2}
37 1−4.70T+37T2 1 - 4.70T + 37T^{2}
41 1−4.47T+41T2 1 - 4.47T + 41T^{2}
43 1+8T+43T2 1 + 8T + 43T^{2}
47 1−7.47T+47T2 1 - 7.47T + 47T^{2}
53 1−7.47T+53T2 1 - 7.47T + 53T^{2}
59 1−1.47T+59T2 1 - 1.47T + 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+2.70T+73T2 1 + 2.70T + 73T^{2}
79 1+2.70T+79T2 1 + 2.70T + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+2.23T+89T2 1 + 2.23T + 89T^{2}
97 1−9.41T+97T2 1 - 9.41T + 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

−7.69629757132774707611651376452, −7.17955611398549923853908406109, −6.62305805131218474681205173489, −5.81398570354841287006827766299, −5.04922240682503243801843755772, −4.40802129489538029928248668667, −3.79572697239028377233237807751, −3.16810263603139382652102846667, −2.31518502414556948623382678579, −1.05093587368298752338815604664, 1.05093587368298752338815604664, 2.31518502414556948623382678579, 3.16810263603139382652102846667, 3.79572697239028377233237807751, 4.40802129489538029928248668667, 5.04922240682503243801843755772, 5.81398570354841287006827766299, 6.62305805131218474681205173489, 7.17955611398549923853908406109, 7.69629757132774707611651376452

Graph of the ZZ-function along the critical line