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Properties

Label 2-4925-1.1-c1-0-10
Degree 22
Conductor 49254925
Sign 11
Analytic cond. 39.326339.3263
Root an. cond. 6.271076.27107
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  − 2.64·2-s + 0.795·3-s + 5.02·4-s − 2.10·6-s − 2.31·7-s − 8.00·8-s − 2.36·9-s − 4.06·11-s + 3.99·12-s − 0.795·13-s + 6.13·14-s + 11.1·16-s − 5.31·17-s + 6.27·18-s − 2.77·19-s − 1.83·21-s + 10.7·22-s − 8.05·23-s − 6.36·24-s + 2.10·26-s − 4.26·27-s − 11.6·28-s + 7.71·29-s + 4.04·31-s − 13.5·32-s − 3.23·33-s + 14.0·34-s + ⋯
L(s)  = 1  − 1.87·2-s + 0.459·3-s + 2.51·4-s − 0.860·6-s − 0.874·7-s − 2.83·8-s − 0.789·9-s − 1.22·11-s + 1.15·12-s − 0.220·13-s + 1.63·14-s + 2.79·16-s − 1.28·17-s + 1.47·18-s − 0.636·19-s − 0.401·21-s + 2.29·22-s − 1.68·23-s − 1.29·24-s + 0.413·26-s − 0.821·27-s − 2.19·28-s + 1.43·29-s + 0.726·31-s − 2.40·32-s − 0.562·33-s + 2.41·34-s + ⋯

Functional equation

Λ(s)=(4925s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4925s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 49254925    =    52⋅1975^{2} \cdot 197
Sign: 11
Analytic conductor: 39.326339.3263
Root analytic conductor: 6.271076.27107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4925, ( :1/2), 1)(2,\ 4925,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.14269668510.1426966851
L(12)L(\frac12) ≈\approx 0.14269668510.1426966851
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)
bad5 1 1
197 1+T 1 + T
good2 1+2.64T+2T2 1 + 2.64T + 2T^{2}
3 1−0.795T+3T2 1 - 0.795T + 3T^{2}
7 1+2.31T+7T2 1 + 2.31T + 7T^{2}
11 1+4.06T+11T2 1 + 4.06T + 11T^{2}
13 1+0.795T+13T2 1 + 0.795T + 13T^{2}
17 1+5.31T+17T2 1 + 5.31T + 17T^{2}
19 1+2.77T+19T2 1 + 2.77T + 19T^{2}
23 1+8.05T+23T2 1 + 8.05T + 23T^{2}
29 1−7.71T+29T2 1 - 7.71T + 29T^{2}
31 1−4.04T+31T2 1 - 4.04T + 31T^{2}
37 1−6.57T+37T2 1 - 6.57T + 37T^{2}
41 1+10.0T+41T2 1 + 10.0T + 41T^{2}
43 1+7.11T+43T2 1 + 7.11T + 43T^{2}
47 1−2.38T+47T2 1 - 2.38T + 47T^{2}
53 1+13.5T+53T2 1 + 13.5T + 53T^{2}
59 1−11.8T+59T2 1 - 11.8T + 59T^{2}
61 1−9.52T+61T2 1 - 9.52T + 61T^{2}
67 1+2.42T+67T2 1 + 2.42T + 67T^{2}
71 1+8.31T+71T2 1 + 8.31T + 71T^{2}
73 1+16.3T+73T2 1 + 16.3T + 73T^{2}
79 1+13.0T+79T2 1 + 13.0T + 79T^{2}
83 1+2.69T+83T2 1 + 2.69T + 83T^{2}
89 1+3.57T+89T2 1 + 3.57T + 89T^{2}
97 1+5.43T+97T2 1 + 5.43T + 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

−8.440676507145179809697855778806, −7.927888572077619365719329143166, −6.99353759662930450860840721712, −6.40420037495449815775459067689, −5.79760316355393600097993593646, −4.50054266563053600645886641969, −3.12410297869396013528651558164, −2.61050999878008539025472366918, −1.89453924246241008216507570086, −0.24530493089577228690068579332, 0.24530493089577228690068579332, 1.89453924246241008216507570086, 2.61050999878008539025472366918, 3.12410297869396013528651558164, 4.50054266563053600645886641969, 5.79760316355393600097993593646, 6.40420037495449815775459067689, 6.99353759662930450860840721712, 7.927888572077619365719329143166, 8.440676507145179809697855778806

Graph of the ZZ-function along the critical line