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Properties

Label 2-2523-1.1-c1-0-31
Degree 22
Conductor 25232523
Sign 11
Analytic cond. 20.146220.1462
Root an. cond. 4.488454.48845
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  − 0.144·2-s + 3-s − 1.97·4-s + 1.55·5-s − 0.144·6-s − 1.52·7-s + 0.574·8-s + 9-s − 0.224·10-s − 0.767·11-s − 1.97·12-s − 2.61·13-s + 0.220·14-s + 1.55·15-s + 3.87·16-s − 3.51·17-s − 0.144·18-s + 5.24·19-s − 3.07·20-s − 1.52·21-s + 0.110·22-s + 8.49·23-s + 0.574·24-s − 2.58·25-s + 0.377·26-s + 27-s + 3.02·28-s + ⋯
L(s)  = 1  − 0.102·2-s + 0.577·3-s − 0.989·4-s + 0.695·5-s − 0.0589·6-s − 0.577·7-s + 0.203·8-s + 0.333·9-s − 0.0709·10-s − 0.231·11-s − 0.571·12-s − 0.726·13-s + 0.0589·14-s + 0.401·15-s + 0.968·16-s − 0.852·17-s − 0.0340·18-s + 1.20·19-s − 0.687·20-s − 0.333·21-s + 0.0236·22-s + 1.77·23-s + 0.117·24-s − 0.516·25-s + 0.0741·26-s + 0.192·27-s + 0.571·28-s + ⋯

Functional equation

Λ(s)=(2523s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2523s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25232523    =    3⋅2923 \cdot 29^{2}
Sign: 11
Analytic conductor: 20.146220.1462
Root analytic conductor: 4.488454.48845
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2523, ( :1/2), 1)(2,\ 2523,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6695722831.669572283
L(12)L(\frac12) ≈\approx 1.6695722831.669572283
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−T 1 - T
29 1 1
good2 1+0.144T+2T2 1 + 0.144T + 2T^{2}
5 1−1.55T+5T2 1 - 1.55T + 5T^{2}
7 1+1.52T+7T2 1 + 1.52T + 7T^{2}
11 1+0.767T+11T2 1 + 0.767T + 11T^{2}
13 1+2.61T+13T2 1 + 2.61T + 13T^{2}
17 1+3.51T+17T2 1 + 3.51T + 17T^{2}
19 1−5.24T+19T2 1 - 5.24T + 19T^{2}
23 1−8.49T+23T2 1 - 8.49T + 23T^{2}
31 1+1.41T+31T2 1 + 1.41T + 31T^{2}
37 1+3.00T+37T2 1 + 3.00T + 37T^{2}
41 1−5.79T+41T2 1 - 5.79T + 41T^{2}
43 1−4.29T+43T2 1 - 4.29T + 43T^{2}
47 1−1.77T+47T2 1 - 1.77T + 47T^{2}
53 1−6.77T+53T2 1 - 6.77T + 53T^{2}
59 1−14.9T+59T2 1 - 14.9T + 59T^{2}
61 1−13.7T+61T2 1 - 13.7T + 61T^{2}
67 1+7.29T+67T2 1 + 7.29T + 67T^{2}
71 1+4.81T+71T2 1 + 4.81T + 71T^{2}
73 1−9.19T+73T2 1 - 9.19T + 73T^{2}
79 1−0.545T+79T2 1 - 0.545T + 79T^{2}
83 1−11.6T+83T2 1 - 11.6T + 83T^{2}
89 1−13.7T+89T2 1 - 13.7T + 89T^{2}
97 1+1.77T+97T2 1 + 1.77T + 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

−9.210375907212062813786111151244, −8.301741840946682430188224384520, −7.41659548121123750374721641780, −6.76541088176346544800293683128, −5.57395986418026934343744725035, −5.05689275118751751026067913563, −4.05829933450750100228103494494, −3.14610368748903989360520842296, −2.24123475052888654306924849141, −0.818623466191123082196960760735, 0.818623466191123082196960760735, 2.24123475052888654306924849141, 3.14610368748903989360520842296, 4.05829933450750100228103494494, 5.05689275118751751026067913563, 5.57395986418026934343744725035, 6.76541088176346544800293683128, 7.41659548121123750374721641780, 8.301741840946682430188224384520, 9.210375907212062813786111151244

Graph of the ZZ-function along the critical line