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Properties

Label 2-45e2-1.1-c1-0-26
Degree 22
Conductor 20252025
Sign 11
Analytic cond. 16.169716.1697
Root an. cond. 4.021154.02115
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  + 0.732·2-s − 1.46·4-s + 4.73·7-s − 2.53·8-s + 5.73·11-s − 1.46·13-s + 3.46·14-s + 1.07·16-s − 2.73·17-s + 4.46·19-s + 4.19·22-s − 3.46·23-s − 1.07·26-s − 6.92·28-s − 3.19·29-s − 3·31-s + 5.85·32-s − 2·34-s + 2.73·37-s + 3.26·38-s + 7.19·41-s − 0.196·43-s − 8.39·44-s − 2.53·46-s − 8.73·47-s + 15.3·49-s + 2.14·52-s + ⋯
L(s)  = 1  + 0.517·2-s − 0.732·4-s + 1.78·7-s − 0.896·8-s + 1.72·11-s − 0.406·13-s + 0.925·14-s + 0.267·16-s − 0.662·17-s + 1.02·19-s + 0.894·22-s − 0.722·23-s − 0.210·26-s − 1.30·28-s − 0.593·29-s − 0.538·31-s + 1.03·32-s − 0.342·34-s + 0.449·37-s + 0.530·38-s + 1.12·41-s − 0.0299·43-s − 1.26·44-s − 0.373·46-s − 1.27·47-s + 2.19·49-s + 0.297·52-s + ⋯

Functional equation

Λ(s)=(2025s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2025s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20252025    =    34⋅523^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 16.169716.1697
Root analytic conductor: 4.021154.02115
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2025, ( :1/2), 1)(2,\ 2025,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4431391262.443139126
L(12)L(\frac12) ≈\approx 2.4431391262.443139126
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 1
5 1 1
good2 1−0.732T+2T2 1 - 0.732T + 2T^{2}
7 1−4.73T+7T2 1 - 4.73T + 7T^{2}
11 1−5.73T+11T2 1 - 5.73T + 11T^{2}
13 1+1.46T+13T2 1 + 1.46T + 13T^{2}
17 1+2.73T+17T2 1 + 2.73T + 17T^{2}
19 1−4.46T+19T2 1 - 4.46T + 19T^{2}
23 1+3.46T+23T2 1 + 3.46T + 23T^{2}
29 1+3.19T+29T2 1 + 3.19T + 29T^{2}
31 1+3T+31T2 1 + 3T + 31T^{2}
37 1−2.73T+37T2 1 - 2.73T + 37T^{2}
41 1−7.19T+41T2 1 - 7.19T + 41T^{2}
43 1+0.196T+43T2 1 + 0.196T + 43T^{2}
47 1+8.73T+47T2 1 + 8.73T + 47T^{2}
53 1−6.73T+53T2 1 - 6.73T + 53T^{2}
59 1−8.26T+59T2 1 - 8.26T + 59T^{2}
61 1−4T+61T2 1 - 4T + 61T^{2}
67 1+3.46T+67T2 1 + 3.46T + 67T^{2}
71 1−3.73T+71T2 1 - 3.73T + 71T^{2}
73 1−7.66T+73T2 1 - 7.66T + 73T^{2}
79 1−15.4T+79T2 1 - 15.4T + 79T^{2}
83 1−2.19T+83T2 1 - 2.19T + 83T^{2}
89 1+5.19T+89T2 1 + 5.19T + 89T^{2}
97 1−9.66T+97T2 1 - 9.66T + 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

−9.178420613225847899306843651419, −8.377417815955795356086287445853, −7.70708037315806450811749373635, −6.72992373634061862250813707979, −5.71049004740771746746258154533, −5.01455882449624897994248053546, −4.26519752515558774275624372848, −3.68340778322905640421676991768, −2.14719958318733903217247076994, −1.05434842431233943681872318850, 1.05434842431233943681872318850, 2.14719958318733903217247076994, 3.68340778322905640421676991768, 4.26519752515558774275624372848, 5.01455882449624897994248053546, 5.71049004740771746746258154533, 6.72992373634061862250813707979, 7.70708037315806450811749373635, 8.377417815955795356086287445853, 9.178420613225847899306843651419

Graph of the ZZ-function along the critical line