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Properties

Label 2-5733-1.1-c1-0-119
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  + 2.44·2-s + 3.99·4-s + 0.910·5-s + 4.87·8-s + 2.22·10-s − 3.67·11-s − 13-s + 3.95·16-s + 7.18·17-s + 1.97·19-s + 3.63·20-s − 9.00·22-s + 0.596·23-s − 4.17·25-s − 2.44·26-s + 3.64·29-s + 7.08·31-s − 0.0786·32-s + 17.5·34-s + 0.710·37-s + 4.84·38-s + 4.43·40-s + 5.27·41-s + 11.0·43-s − 14.6·44-s + 1.46·46-s + 12.1·47-s + ⋯
L(s)  = 1  + 1.73·2-s + 1.99·4-s + 0.407·5-s + 1.72·8-s + 0.704·10-s − 1.10·11-s − 0.277·13-s + 0.987·16-s + 1.74·17-s + 0.453·19-s + 0.812·20-s − 1.91·22-s + 0.124·23-s − 0.834·25-s − 0.480·26-s + 0.677·29-s + 1.27·31-s − 0.0139·32-s + 3.01·34-s + 0.116·37-s + 0.785·38-s + 0.701·40-s + 0.823·41-s + 1.68·43-s − 2.21·44-s + 0.215·46-s + 1.76·47-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 6.5108292536.510829253
L(12)L(\frac12) ≈\approx 6.5108292536.510829253
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 1−2.44T+2T2 1 - 2.44T + 2T^{2}
5 1−0.910T+5T2 1 - 0.910T + 5T^{2}
11 1+3.67T+11T2 1 + 3.67T + 11T^{2}
17 1−7.18T+17T2 1 - 7.18T + 17T^{2}
19 1−1.97T+19T2 1 - 1.97T + 19T^{2}
23 1−0.596T+23T2 1 - 0.596T + 23T^{2}
29 1−3.64T+29T2 1 - 3.64T + 29T^{2}
31 1−7.08T+31T2 1 - 7.08T + 31T^{2}
37 1−0.710T+37T2 1 - 0.710T + 37T^{2}
41 1−5.27T+41T2 1 - 5.27T + 41T^{2}
43 1−11.0T+43T2 1 - 11.0T + 43T^{2}
47 1−12.1T+47T2 1 - 12.1T + 47T^{2}
53 1−11.4T+53T2 1 - 11.4T + 53T^{2}
59 1−9.58T+59T2 1 - 9.58T + 59T^{2}
61 1+6.98T+61T2 1 + 6.98T + 61T^{2}
67 1−1.22T+67T2 1 - 1.22T + 67T^{2}
71 1+11.3T+71T2 1 + 11.3T + 71T^{2}
73 1+6.53T+73T2 1 + 6.53T + 73T^{2}
79 1+11.5T+79T2 1 + 11.5T + 79T^{2}
83 1−7.16T+83T2 1 - 7.16T + 83T^{2}
89 1−12.8T+89T2 1 - 12.8T + 89T^{2}
97 1+9.09T+97T2 1 + 9.09T + 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

−7.55672720972543977570327098006, −7.50611141007961290864308855772, −6.33199337022978923502086042326, −5.61763954418002696913496740009, −5.43029422686498172875242121096, −4.48267677594962619251184766592, −3.80712864267146896367255347370, −2.76088185299542786441067534025, −2.51122998091485589399661627935, −1.08979065815532064532752235635, 1.08979065815532064532752235635, 2.51122998091485589399661627935, 2.76088185299542786441067534025, 3.80712864267146896367255347370, 4.48267677594962619251184766592, 5.43029422686498172875242121096, 5.61763954418002696913496740009, 6.33199337022978923502086042326, 7.50611141007961290864308855772, 7.55672720972543977570327098006

Graph of the ZZ-function along the critical line