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Properties

Label 2-538-1.1-c3-0-23
Degree 22
Conductor 538538
Sign 11
Analytic cond. 31.743031.7430
Root an. cond. 5.634095.63409
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 6.41·3-s + 4·4-s + 6.01·5-s − 12.8·6-s + 26.2·7-s + 8·8-s + 14.1·9-s + 12.0·10-s + 69.8·11-s − 25.6·12-s − 28.9·13-s + 52.5·14-s − 38.6·15-s + 16·16-s − 15.4·17-s + 28.3·18-s − 84.7·19-s + 24.0·20-s − 168.·21-s + 139.·22-s + 35.7·23-s − 51.3·24-s − 88.7·25-s − 57.9·26-s + 82.2·27-s + 105.·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.23·3-s + 0.5·4-s + 0.538·5-s − 0.873·6-s + 1.41·7-s + 0.353·8-s + 0.525·9-s + 0.380·10-s + 1.91·11-s − 0.617·12-s − 0.618·13-s + 1.00·14-s − 0.664·15-s + 0.250·16-s − 0.221·17-s + 0.371·18-s − 1.02·19-s + 0.269·20-s − 1.75·21-s + 1.35·22-s + 0.324·23-s − 0.436·24-s − 0.710·25-s − 0.437·26-s + 0.585·27-s + 0.709·28-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 11
Analytic conductor: 31.743031.7430
Root analytic conductor: 5.634095.63409
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 538, ( :3/2), 1)(2,\ 538,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.8315741262.831574126
L(12)L(\frac12) ≈\approx 2.8315741262.831574126
L(52)L(\frac{5}{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 1−2T 1 - 2T
269 1−269T 1 - 269T
good3 1+6.41T+27T2 1 + 6.41T + 27T^{2}
5 1−6.01T+125T2 1 - 6.01T + 125T^{2}
7 1−26.2T+343T2 1 - 26.2T + 343T^{2}
11 1−69.8T+1.33e3T2 1 - 69.8T + 1.33e3T^{2}
13 1+28.9T+2.19e3T2 1 + 28.9T + 2.19e3T^{2}
17 1+15.4T+4.91e3T2 1 + 15.4T + 4.91e3T^{2}
19 1+84.7T+6.85e3T2 1 + 84.7T + 6.85e3T^{2}
23 1−35.7T+1.21e4T2 1 - 35.7T + 1.21e4T^{2}
29 1+95.2T+2.43e4T2 1 + 95.2T + 2.43e4T^{2}
31 1−189.T+2.97e4T2 1 - 189.T + 2.97e4T^{2}
37 1−76.9T+5.06e4T2 1 - 76.9T + 5.06e4T^{2}
41 1−387.T+6.89e4T2 1 - 387.T + 6.89e4T^{2}
43 1−562.T+7.95e4T2 1 - 562.T + 7.95e4T^{2}
47 1+469.T+1.03e5T2 1 + 469.T + 1.03e5T^{2}
53 1−424.T+1.48e5T2 1 - 424.T + 1.48e5T^{2}
59 1+490.T+2.05e5T2 1 + 490.T + 2.05e5T^{2}
61 1−564.T+2.26e5T2 1 - 564.T + 2.26e5T^{2}
67 1−111.T+3.00e5T2 1 - 111.T + 3.00e5T^{2}
71 1+717.T+3.57e5T2 1 + 717.T + 3.57e5T^{2}
73 1−328.T+3.89e5T2 1 - 328.T + 3.89e5T^{2}
79 1−557.T+4.93e5T2 1 - 557.T + 4.93e5T^{2}
83 1+873.T+5.71e5T2 1 + 873.T + 5.71e5T^{2}
89 1−1.05e3T+7.04e5T2 1 - 1.05e3T + 7.04e5T^{2}
97 1+1.30e3T+9.12e5T2 1 + 1.30e3T + 9.12e5T^{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

−10.88144685885327027391919688022, −9.713784300066406088737294012044, −8.651883210101215581855232787263, −7.40139580584963469607197693867, −6.37343046911279419029154373859, −5.82332603079908902875583430299, −4.74989835915046747677367145062, −4.15335386323101858838104761138, −2.17992625302447889317828734785, −1.06770949401608027107714181934, 1.06770949401608027107714181934, 2.17992625302447889317828734785, 4.15335386323101858838104761138, 4.74989835915046747677367145062, 5.82332603079908902875583430299, 6.37343046911279419029154373859, 7.40139580584963469607197693867, 8.651883210101215581855232787263, 9.713784300066406088737294012044, 10.88144685885327027391919688022

Graph of the ZZ-function along the critical line