↑

Properties

Label 2-538-1.1-c1-0-7
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.34·3-s + 4-s + 3.90·5-s − 1.34·6-s − 0.487·7-s − 8-s − 1.19·9-s − 3.90·10-s + 4.04·11-s + 1.34·12-s + 3.53·13-s + 0.487·14-s + 5.24·15-s + 16-s − 6.37·17-s + 1.19·18-s + 0.975·19-s + 3.90·20-s − 0.655·21-s − 4.04·22-s + 0.750·23-s − 1.34·24-s + 10.2·25-s − 3.53·26-s − 5.63·27-s − 0.487·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.776·3-s + 0.5·4-s + 1.74·5-s − 0.548·6-s − 0.184·7-s − 0.353·8-s − 0.397·9-s − 1.23·10-s + 1.22·11-s + 0.388·12-s + 0.981·13-s + 0.130·14-s + 1.35·15-s + 0.250·16-s − 1.54·17-s + 0.281·18-s + 0.223·19-s + 0.873·20-s − 0.143·21-s − 0.863·22-s + 0.156·23-s − 0.274·24-s + 2.05·25-s − 0.693·26-s − 1.08·27-s − 0.0922·28-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 1.7729183891.772918389
L(12)L(\frac12) ≈\approx 1.7729183891.772918389
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 1−1.34T+3T2 1 - 1.34T + 3T^{2}
5 1−3.90T+5T2 1 - 3.90T + 5T^{2}
7 1+0.487T+7T2 1 + 0.487T + 7T^{2}
11 1−4.04T+11T2 1 - 4.04T + 11T^{2}
13 1−3.53T+13T2 1 - 3.53T + 13T^{2}
17 1+6.37T+17T2 1 + 6.37T + 17T^{2}
19 1−0.975T+19T2 1 - 0.975T + 19T^{2}
23 1−0.750T+23T2 1 - 0.750T + 23T^{2}
29 1+4.08T+29T2 1 + 4.08T + 29T^{2}
31 1+7.86T+31T2 1 + 7.86T + 31T^{2}
37 1+4.22T+37T2 1 + 4.22T + 37T^{2}
41 1+2.07T+41T2 1 + 2.07T + 41T^{2}
43 1−10.7T+43T2 1 - 10.7T + 43T^{2}
47 1−9.31T+47T2 1 - 9.31T + 47T^{2}
53 1−11.1T+53T2 1 - 11.1T + 53T^{2}
59 1+2.98T+59T2 1 + 2.98T + 59T^{2}
61 1+7.53T+61T2 1 + 7.53T + 61T^{2}
67 1−3.51T+67T2 1 - 3.51T + 67T^{2}
71 1−14.1T+71T2 1 - 14.1T + 71T^{2}
73 1+9.75T+73T2 1 + 9.75T + 73T^{2}
79 1−2.29T+79T2 1 - 2.29T + 79T^{2}
83 1+17.1T+83T2 1 + 17.1T + 83T^{2}
89 1−1.86T+89T2 1 - 1.86T + 89T^{2}
97 1+9.63T+97T2 1 + 9.63T + 97T^{2}
show more
show less
   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

−10.70647176915355308046833031336, −9.565074878449781993456058801529, −9.009815312926693731781485797971, −8.727348526109066696294612888373, −7.18430664689741418114004597618, −6.30196371455321299080964928772, −5.60533450717780083541617327027, −3.80936701963792566063123451023, −2.48415845756228362038085488963, −1.58602000741028475136762805962, 1.58602000741028475136762805962, 2.48415845756228362038085488963, 3.80936701963792566063123451023, 5.60533450717780083541617327027, 6.30196371455321299080964928772, 7.18430664689741418114004597618, 8.727348526109066696294612888373, 9.009815312926693731781485797971, 9.565074878449781993456058801529, 10.70647176915355308046833031336

Graph of the ZZ-function along the critical line