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Properties

Label 2-538-1.1-c1-0-12
Degree 22
Conductor 538538
Sign 11
Analytic cond. 4.295954.29595
Root an. cond. 2.072662.07266
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 + 1.65·3-s + 4-s + 0.725·5-s + 1.65·6-s + 1.29·7-s + 8-s − 0.269·9-s + 0.725·10-s − 3.71·11-s + 1.65·12-s + 2.30·13-s + 1.29·14-s + 1.19·15-s + 16-s + 6.32·17-s − 0.269·18-s + 0.0212·19-s + 0.725·20-s + 2.14·21-s − 3.71·22-s − 1.30·23-s + 1.65·24-s − 4.47·25-s + 2.30·26-s − 5.40·27-s + 1.29·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.954·3-s + 0.5·4-s + 0.324·5-s + 0.674·6-s + 0.491·7-s + 0.353·8-s − 0.0896·9-s + 0.229·10-s − 1.12·11-s + 0.477·12-s + 0.640·13-s + 0.347·14-s + 0.309·15-s + 0.250·16-s + 1.53·17-s − 0.0634·18-s + 0.00487·19-s + 0.162·20-s + 0.468·21-s − 0.792·22-s − 0.272·23-s + 0.337·24-s − 0.894·25-s + 0.452·26-s − 1.03·27-s + 0.245·28-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 11
Analytic conductor: 4.295954.29595
Root analytic conductor: 2.072662.07266
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 538, ( :1/2), 1)(2,\ 538,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0393920603.039392060
L(12)L(\frac12) ≈\approx 3.0393920603.039392060
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
269 1+T 1 + T
good3 1−1.65T+3T2 1 - 1.65T + 3T^{2}
5 1−0.725T+5T2 1 - 0.725T + 5T^{2}
7 1−1.29T+7T2 1 - 1.29T + 7T^{2}
11 1+3.71T+11T2 1 + 3.71T + 11T^{2}
13 1−2.30T+13T2 1 - 2.30T + 13T^{2}
17 1−6.32T+17T2 1 - 6.32T + 17T^{2}
19 1−0.0212T+19T2 1 - 0.0212T + 19T^{2}
23 1+1.30T+23T2 1 + 1.30T + 23T^{2}
29 1+2.06T+29T2 1 + 2.06T + 29T^{2}
31 1+5.82T+31T2 1 + 5.82T + 31T^{2}
37 1+7.25T+37T2 1 + 7.25T + 37T^{2}
41 1−8.77T+41T2 1 - 8.77T + 41T^{2}
43 1−4.52T+43T2 1 - 4.52T + 43T^{2}
47 1−3.85T+47T2 1 - 3.85T + 47T^{2}
53 1+6.86T+53T2 1 + 6.86T + 53T^{2}
59 1−7.84T+59T2 1 - 7.84T + 59T^{2}
61 1−3.84T+61T2 1 - 3.84T + 61T^{2}
67 1+9.31T+67T2 1 + 9.31T + 67T^{2}
71 1+0.643T+71T2 1 + 0.643T + 71T^{2}
73 1+2.97T+73T2 1 + 2.97T + 73T^{2}
79 1−0.418T+79T2 1 - 0.418T + 79T^{2}
83 1+15.6T+83T2 1 + 15.6T + 83T^{2}
89 1−10.3T+89T2 1 - 10.3T + 89T^{2}
97 1−5.07T+97T2 1 - 5.07T + 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

−10.85470600500216461815499180265, −9.991636738218995253077395919556, −8.972093888940116060710189778839, −7.955089796853058772826087668280, −7.50622136560456556881854933257, −5.89214041651080648345770327155, −5.33500341220536760231560600725, −3.90074527201998946004366696834, −2.98345055186004453631441149281, −1.86331627140814674766273129677, 1.86331627140814674766273129677, 2.98345055186004453631441149281, 3.90074527201998946004366696834, 5.33500341220536760231560600725, 5.89214041651080648345770327155, 7.50622136560456556881854933257, 7.955089796853058772826087668280, 8.972093888940116060710189778839, 9.991636738218995253077395919556, 10.85470600500216461815499180265

Graph of the ZZ-function along the critical line