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Properties

Label 2-538-1.1-c1-0-19
Degree 22
Conductor 538538
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.61·3-s + 4-s − 1.38·5-s − 1.61·6-s + 0.236·7-s + 8-s − 0.381·9-s − 1.38·10-s − 4.23·11-s − 1.61·12-s − 13-s + 0.236·14-s + 2.23·15-s + 16-s − 4.23·17-s − 0.381·18-s + 2·19-s − 1.38·20-s − 0.381·21-s − 4.23·22-s − 3.76·23-s − 1.61·24-s − 3.09·25-s − 26-s + 5.47·27-s + 0.236·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.934·3-s + 0.5·4-s − 0.618·5-s − 0.660·6-s + 0.0892·7-s + 0.353·8-s − 0.127·9-s − 0.437·10-s − 1.27·11-s − 0.467·12-s − 0.277·13-s + 0.0630·14-s + 0.577·15-s + 0.250·16-s − 1.02·17-s − 0.0900·18-s + 0.458·19-s − 0.309·20-s − 0.0833·21-s − 0.903·22-s − 0.784·23-s − 0.330·24-s − 0.618·25-s − 0.196·26-s + 1.05·27-s + 0.0446·28-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: −1-1
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: 11
Selberg data: (2, 538, ( :1/2), −1)(2,\ 538,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+1.61T+3T2 1 + 1.61T + 3T^{2}
5 1+1.38T+5T2 1 + 1.38T + 5T^{2}
7 1−0.236T+7T2 1 - 0.236T + 7T^{2}
11 1+4.23T+11T2 1 + 4.23T + 11T^{2}
13 1+T+13T2 1 + T + 13T^{2}
17 1+4.23T+17T2 1 + 4.23T + 17T^{2}
19 1−2T+19T2 1 - 2T + 19T^{2}
23 1+3.76T+23T2 1 + 3.76T + 23T^{2}
29 1+5.85T+29T2 1 + 5.85T + 29T^{2}
31 1+1.47T+31T2 1 + 1.47T + 31T^{2}
37 1−4.70T+37T2 1 - 4.70T + 37T^{2}
41 1+5.70T+41T2 1 + 5.70T + 41T^{2}
43 1+2.23T+43T2 1 + 2.23T + 43T^{2}
47 1+0.145T+47T2 1 + 0.145T + 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1−13.5T+59T2 1 - 13.5T + 59T^{2}
61 1−3.47T+61T2 1 - 3.47T + 61T^{2}
67 1−6.23T+67T2 1 - 6.23T + 67T^{2}
71 1+3.70T+71T2 1 + 3.70T + 71T^{2}
73 1−2.70T+73T2 1 - 2.70T + 73T^{2}
79 1−10.7T+79T2 1 - 10.7T + 79T^{2}
83 1+0.763T+83T2 1 + 0.763T + 83T^{2}
89 1−14.7T+89T2 1 - 14.7T + 89T^{2}
97 1+18.6T+97T2 1 + 18.6T + 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

−10.75550022492265920355365148009, −9.779310370633365036992065654641, −8.340542059575305887345664585683, −7.55313253354201825755994795926, −6.52286239153267665937487243535, −5.53522707827406947590541907383, −4.88465287099634814764121847312, −3.72426930725569106217054384489, −2.34630354444356126931490073198, 0, 2.34630354444356126931490073198, 3.72426930725569106217054384489, 4.88465287099634814764121847312, 5.53522707827406947590541907383, 6.52286239153267665937487243535, 7.55313253354201825755994795926, 8.340542059575305887345664585683, 9.779310370633365036992065654641, 10.75550022492265920355365148009

Graph of the ZZ-function along the critical line