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Properties

Label 2-95e2-1.1-c1-0-511
Degree 22
Conductor 90259025
Sign −1-1
Analytic cond. 72.064972.0649
Root an. cond. 8.489108.48910
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.90·2-s + 1.90·3-s + 1.61·4-s + 3.61·6-s + 4.23·7-s − 0.726·8-s + 0.618·9-s − 5.85·11-s + 3.07·12-s − 3.07·13-s + 8.05·14-s − 4.61·16-s − 5.23·17-s + 1.17·18-s + 8.05·21-s − 11.1·22-s − 4.09·23-s − 1.38·24-s − 5.85·26-s − 4.53·27-s + 6.85·28-s − 2.80·29-s + 1.90·31-s − 7.33·32-s − 11.1·33-s − 9.95·34-s + 0.999·36-s + ⋯
L(s)  = 1  + 1.34·2-s + 1.09·3-s + 0.809·4-s + 1.47·6-s + 1.60·7-s − 0.256·8-s + 0.206·9-s − 1.76·11-s + 0.888·12-s − 0.853·13-s + 2.15·14-s − 1.15·16-s − 1.26·17-s + 0.277·18-s + 1.75·21-s − 2.37·22-s − 0.852·23-s − 0.282·24-s − 1.14·26-s − 0.871·27-s + 1.29·28-s − 0.519·29-s + 0.341·31-s − 1.29·32-s − 1.93·33-s − 1.70·34-s + 0.166·36-s + ⋯

Functional equation

Λ(s)=(9025s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9025s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 90259025    =    52⋅1925^{2} \cdot 19^{2}
Sign: −1-1
Analytic conductor: 72.064972.0649
Root analytic conductor: 8.489108.48910
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9025, ( :1/2), −1)(2,\ 9025,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
19 1 1
good2 1−1.90T+2T2 1 - 1.90T + 2T^{2}
3 1−1.90T+3T2 1 - 1.90T + 3T^{2}
7 1−4.23T+7T2 1 - 4.23T + 7T^{2}
11 1+5.85T+11T2 1 + 5.85T + 11T^{2}
13 1+3.07T+13T2 1 + 3.07T + 13T^{2}
17 1+5.23T+17T2 1 + 5.23T + 17T^{2}
23 1+4.09T+23T2 1 + 4.09T + 23T^{2}
29 1+2.80T+29T2 1 + 2.80T + 29T^{2}
31 1−1.90T+31T2 1 - 1.90T + 31T^{2}
37 1−2.80T+37T2 1 - 2.80T + 37T^{2}
41 1+6.88T+41T2 1 + 6.88T + 41T^{2}
43 1+0.381T+43T2 1 + 0.381T + 43T^{2}
47 1−1.47T+47T2 1 - 1.47T + 47T^{2}
53 1+11.1T+53T2 1 + 11.1T + 53T^{2}
59 1−14.0T+59T2 1 - 14.0T + 59T^{2}
61 1−3.94T+61T2 1 - 3.94T + 61T^{2}
67 1+5.98T+67T2 1 + 5.98T + 67T^{2}
71 1−0.171T+71T2 1 - 0.171T + 71T^{2}
73 1−T+73T2 1 - T + 73T^{2}
79 1+5.25T+79T2 1 + 5.25T + 79T^{2}
83 1−8.76T+83T2 1 - 8.76T + 83T^{2}
89 1−7.77T+89T2 1 - 7.77T + 89T^{2}
97 1−2.62T+97T2 1 - 2.62T + 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

−7.61818666960757053362030214104, −6.65401389506258334832666580343, −5.66734294314587286774730887540, −5.09611144754851109528460127156, −4.65735883141102259682088182730, −3.95994281196290031332710496106, −3.03898527746209755051834103371, −2.29186905702727473181062185612, −2.04356528642148507974529970904, 0, 2.04356528642148507974529970904, 2.29186905702727473181062185612, 3.03898527746209755051834103371, 3.95994281196290031332710496106, 4.65735883141102259682088182730, 5.09611144754851109528460127156, 5.66734294314587286774730887540, 6.65401389506258334832666580343, 7.61818666960757053362030214104

Graph of the ZZ-function along the critical line