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Properties

Label 2-95-1.1-c1-0-3
Degree 22
Conductor 9595
Sign 11
Analytic cond. 0.7585780.758578
Root an. cond. 0.8709640.870964
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  + 0.816·2-s + 1.53·3-s − 1.33·4-s − 5-s + 1.25·6-s + 5.03·7-s − 2.72·8-s − 0.633·9-s − 0.816·10-s − 3.03·11-s − 2.05·12-s − 4.57·13-s + 4.11·14-s − 1.53·15-s + 0.443·16-s − 1.07·17-s − 0.517·18-s + 19-s + 1.33·20-s + 7.74·21-s − 2.47·22-s + 4.11·23-s − 4.18·24-s + 25-s − 3.73·26-s − 5.58·27-s − 6.71·28-s + ⋯
L(s)  = 1  + 0.577·2-s + 0.888·3-s − 0.666·4-s − 0.447·5-s + 0.512·6-s + 1.90·7-s − 0.962·8-s − 0.211·9-s − 0.258·10-s − 0.914·11-s − 0.592·12-s − 1.26·13-s + 1.09·14-s − 0.397·15-s + 0.110·16-s − 0.261·17-s − 0.121·18-s + 0.229·19-s + 0.298·20-s + 1.68·21-s − 0.528·22-s + 0.857·23-s − 0.854·24-s + 0.200·25-s − 0.732·26-s − 1.07·27-s − 1.26·28-s + ⋯

Functional equation

Λ(s)=(95s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(95s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9595    =    5⋅195 \cdot 19
Sign: 11
Analytic conductor: 0.7585780.758578
Root analytic conductor: 0.8709640.870964
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 95, ( :1/2), 1)(2,\ 95,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3756361581.375636158
L(12)L(\frac12) ≈\approx 1.3756361581.375636158
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+T 1 + T
19 1−T 1 - T
good2 1−0.816T+2T2 1 - 0.816T + 2T^{2}
3 1−1.53T+3T2 1 - 1.53T + 3T^{2}
7 1−5.03T+7T2 1 - 5.03T + 7T^{2}
11 1+3.03T+11T2 1 + 3.03T + 11T^{2}
13 1+4.57T+13T2 1 + 4.57T + 13T^{2}
17 1+1.07T+17T2 1 + 1.07T + 17T^{2}
23 1−4.11T+23T2 1 - 4.11T + 23T^{2}
29 1+1.07T+29T2 1 + 1.07T + 29T^{2}
31 1−5.58T+31T2 1 - 5.58T + 31T^{2}
37 1−0.0947T+37T2 1 - 0.0947T + 37T^{2}
41 1−10.6T+41T2 1 - 10.6T + 41T^{2}
43 1−5.03T+43T2 1 - 5.03T + 43T^{2}
47 1+12.2T+47T2 1 + 12.2T + 47T^{2}
53 1+4.09T+53T2 1 + 4.09T + 53T^{2}
59 1+1.39T+59T2 1 + 1.39T + 59T^{2}
61 1+5.69T+61T2 1 + 5.69T + 61T^{2}
67 1−5.28T+67T2 1 - 5.28T + 67T^{2}
71 1+5.67T+71T2 1 + 5.67T + 71T^{2}
73 1−9.07T+73T2 1 - 9.07T + 73T^{2}
79 1+5.39T+79T2 1 + 5.39T + 79T^{2}
83 1−1.95T+83T2 1 - 1.95T + 83T^{2}
89 1+2.18T+89T2 1 + 2.18T + 89T^{2}
97 1+2.16T+97T2 1 + 2.16T + 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

−14.26130826641689113741472137633, −13.18340986649596871940767958234, −12.01852929511958864368013333007, −10.98458192117391506294016079182, −9.389715201950502876245592725518, −8.276465298966013527834539267026, −7.68669710050338327374469583993, −5.27426738456968623481319456299, −4.47077755270593959731533655099, −2.70606734603912392472615532973, 2.70606734603912392472615532973, 4.47077755270593959731533655099, 5.27426738456968623481319456299, 7.68669710050338327374469583993, 8.276465298966013527834539267026, 9.389715201950502876245592725518, 10.98458192117391506294016079182, 12.01852929511958864368013333007, 13.18340986649596871940767958234, 14.26130826641689113741472137633

Graph of the ZZ-function along the critical line