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Properties

Label 2-5733-1.1-c1-0-67
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  − 2.18·2-s + 2.76·4-s + 2.11·5-s − 1.67·8-s − 4.60·10-s + 5.76·11-s − 13-s − 1.87·16-s + 1.64·17-s + 2.67·19-s + 5.83·20-s − 12.5·22-s − 6.42·23-s − 0.545·25-s + 2.18·26-s + 6.04·29-s − 5.12·31-s + 7.45·32-s − 3.58·34-s + 5.74·37-s − 5.83·38-s − 3.53·40-s − 7.14·41-s − 4.47·43-s + 15.9·44-s + 14.0·46-s + 11.7·47-s + ⋯
L(s)  = 1  − 1.54·2-s + 1.38·4-s + 0.943·5-s − 0.591·8-s − 1.45·10-s + 1.73·11-s − 0.277·13-s − 0.469·16-s + 0.397·17-s + 0.612·19-s + 1.30·20-s − 2.68·22-s − 1.33·23-s − 0.109·25-s + 0.428·26-s + 1.12·29-s − 0.919·31-s + 1.31·32-s − 0.614·34-s + 0.944·37-s − 0.945·38-s − 0.558·40-s − 1.11·41-s − 0.681·43-s + 2.40·44-s + 2.06·46-s + 1.71·47-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 1.2475538641.247553864
L(12)L(\frac12) ≈\approx 1.2475538641.247553864
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+2.18T+2T2 1 + 2.18T + 2T^{2}
5 1−2.11T+5T2 1 - 2.11T + 5T^{2}
11 1−5.76T+11T2 1 - 5.76T + 11T^{2}
17 1−1.64T+17T2 1 - 1.64T + 17T^{2}
19 1−2.67T+19T2 1 - 2.67T + 19T^{2}
23 1+6.42T+23T2 1 + 6.42T + 23T^{2}
29 1−6.04T+29T2 1 - 6.04T + 29T^{2}
31 1+5.12T+31T2 1 + 5.12T + 31T^{2}
37 1−5.74T+37T2 1 - 5.74T + 37T^{2}
41 1+7.14T+41T2 1 + 7.14T + 41T^{2}
43 1+4.47T+43T2 1 + 4.47T + 43T^{2}
47 1−11.7T+47T2 1 - 11.7T + 47T^{2}
53 1+3.44T+53T2 1 + 3.44T + 53T^{2}
59 1−13.1T+59T2 1 - 13.1T + 59T^{2}
61 1+6.24T+61T2 1 + 6.24T + 61T^{2}
67 1−7.74T+67T2 1 - 7.74T + 67T^{2}
71 1+13.6T+71T2 1 + 13.6T + 71T^{2}
73 1−15.5T+73T2 1 - 15.5T + 73T^{2}
79 1−1.12T+79T2 1 - 1.12T + 79T^{2}
83 1−4.96T+83T2 1 - 4.96T + 83T^{2}
89 1−1.14T+89T2 1 - 1.14T + 89T^{2}
97 1−6.97T+97T2 1 - 6.97T + 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

−8.267276373221374825379827996146, −7.55915393153350378739075840417, −6.79638666639168389239299985342, −6.27636869942439914555890857291, −5.52298020904386848728986811061, −4.42561997092519223416257278164, −3.52460765254531105911368139347, −2.28382112034212283214981919499, −1.63833109033798898816016151667, −0.796259506961945288164606082058, 0.796259506961945288164606082058, 1.63833109033798898816016151667, 2.28382112034212283214981919499, 3.52460765254531105911368139347, 4.42561997092519223416257278164, 5.52298020904386848728986811061, 6.27636869942439914555890857291, 6.79638666639168389239299985342, 7.55915393153350378739075840417, 8.267276373221374825379827996146

Graph of the ZZ-function along the critical line