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Properties

Label 2-950-1.1-c1-0-10
Degree 22
Conductor 950950
Sign 11
Analytic cond. 7.585787.58578
Root an. cond. 2.754232.75423
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-s + 3.03·3-s + 4-s − 3.03·6-s − 2.46·7-s − 8-s + 6.19·9-s + 0.728·11-s + 3.03·12-s + 6.23·13-s + 2.46·14-s + 16-s − 0.563·17-s − 6.19·18-s − 19-s − 7.49·21-s − 0.728·22-s − 4.63·23-s − 3.03·24-s − 6.23·26-s + 9.70·27-s − 2.46·28-s + 10.2·29-s + 6.06·31-s − 32-s + 2.21·33-s + 0.563·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.75·3-s + 0.5·4-s − 1.23·6-s − 0.933·7-s − 0.353·8-s + 2.06·9-s + 0.219·11-s + 0.875·12-s + 1.72·13-s + 0.660·14-s + 0.250·16-s − 0.136·17-s − 1.46·18-s − 0.229·19-s − 1.63·21-s − 0.155·22-s − 0.966·23-s − 0.619·24-s − 1.22·26-s + 1.86·27-s − 0.466·28-s + 1.89·29-s + 1.08·31-s − 0.176·32-s + 0.384·33-s + 0.0965·34-s + ⋯

Functional equation

Λ(s)=(950s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(950s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 950950    =    2⋅52⋅192 \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 7.585787.58578
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 950, ( :1/2), 1)(2,\ 950,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1043561652.104356165
L(12)L(\frac12) ≈\approx 2.1043561652.104356165
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)
bad2 1+T 1 + T
5 1 1
19 1+T 1 + T
good3 1−3.03T+3T2 1 - 3.03T + 3T^{2}
7 1+2.46T+7T2 1 + 2.46T + 7T^{2}
11 1−0.728T+11T2 1 - 0.728T + 11T^{2}
13 1−6.23T+13T2 1 - 6.23T + 13T^{2}
17 1+0.563T+17T2 1 + 0.563T + 17T^{2}
23 1+4.63T+23T2 1 + 4.63T + 23T^{2}
29 1−10.2T+29T2 1 - 10.2T + 29T^{2}
31 1−6.06T+31T2 1 - 6.06T + 31T^{2}
37 1−5.72T+37T2 1 - 5.72T + 37T^{2}
41 1−4.79T+41T2 1 - 4.79T + 41T^{2}
43 1+8.06T+43T2 1 + 8.06T + 43T^{2}
47 1+8.12T+47T2 1 + 8.12T + 47T^{2}
53 1−1.53T+53T2 1 - 1.53T + 53T^{2}
59 1+5.76T+59T2 1 + 5.76T + 59T^{2}
61 1−10.9T+61T2 1 - 10.9T + 61T^{2}
67 1+12.9T+67T2 1 + 12.9T + 67T^{2}
71 1+4.39T+71T2 1 + 4.39T + 71T^{2}
73 1−4.09T+73T2 1 - 4.09T + 73T^{2}
79 1−15.3T+79T2 1 - 15.3T + 79T^{2}
83 1+7.85T+83T2 1 + 7.85T + 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1+11.0T+97T2 1 + 11.0T + 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

−9.839291150335701093194524665266, −9.105715558906536678433583184720, −8.343265806475094067947076399705, −8.031710433744774355856835215129, −6.71419546895230966046710993368, −6.23488625279696781589515800554, −4.28496907792576111862592985338, −3.39413889247916756875369629856, −2.63166424789398613292078802838, −1.32777611572376336861192585493, 1.32777611572376336861192585493, 2.63166424789398613292078802838, 3.39413889247916756875369629856, 4.28496907792576111862592985338, 6.23488625279696781589515800554, 6.71419546895230966046710993368, 8.031710433744774355856835215129, 8.343265806475094067947076399705, 9.105715558906536678433583184720, 9.839291150335701093194524665266

Graph of the ZZ-function along the critical line