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Properties

Label 2-416-1.1-c3-0-0
Degree 22
Conductor 416416
Sign 11
Analytic cond. 24.544724.5447
Root an. cond. 4.954274.95427
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  − 4.53·3-s − 20.8·5-s + 16.7·7-s − 6.40·9-s − 72.1·11-s − 13·13-s + 94.4·15-s − 21.2·17-s − 66.7·19-s − 76.2·21-s − 149.·23-s + 307.·25-s + 151.·27-s − 81.0·29-s − 177.·31-s + 327.·33-s − 349.·35-s + 353.·37-s + 58.9·39-s + 284.·41-s + 31.8·43-s + 133.·45-s + 112.·47-s − 60.8·49-s + 96.2·51-s + 211.·53-s + 1.50e3·55-s + ⋯
L(s)  = 1  − 0.873·3-s − 1.86·5-s + 0.906·7-s − 0.237·9-s − 1.97·11-s − 0.277·13-s + 1.62·15-s − 0.302·17-s − 0.805·19-s − 0.792·21-s − 1.35·23-s + 2.46·25-s + 1.08·27-s − 0.518·29-s − 1.02·31-s + 1.72·33-s − 1.68·35-s + 1.56·37-s + 0.242·39-s + 1.08·41-s + 0.112·43-s + 0.441·45-s + 0.347·47-s − 0.177·49-s + 0.264·51-s + 0.548·53-s + 3.68·55-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 11
Analytic conductor: 24.544724.5447
Root analytic conductor: 4.954274.95427
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 416, ( :3/2), 1)(2,\ 416,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.29968538850.2996853885
L(12)L(\frac12) ≈\approx 0.29968538850.2996853885
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 1
13 1+13T 1 + 13T
good3 1+4.53T+27T2 1 + 4.53T + 27T^{2}
5 1+20.8T+125T2 1 + 20.8T + 125T^{2}
7 1−16.7T+343T2 1 - 16.7T + 343T^{2}
11 1+72.1T+1.33e3T2 1 + 72.1T + 1.33e3T^{2}
17 1+21.2T+4.91e3T2 1 + 21.2T + 4.91e3T^{2}
19 1+66.7T+6.85e3T2 1 + 66.7T + 6.85e3T^{2}
23 1+149.T+1.21e4T2 1 + 149.T + 1.21e4T^{2}
29 1+81.0T+2.43e4T2 1 + 81.0T + 2.43e4T^{2}
31 1+177.T+2.97e4T2 1 + 177.T + 2.97e4T^{2}
37 1−353.T+5.06e4T2 1 - 353.T + 5.06e4T^{2}
41 1−284.T+6.89e4T2 1 - 284.T + 6.89e4T^{2}
43 1−31.8T+7.95e4T2 1 - 31.8T + 7.95e4T^{2}
47 1−112.T+1.03e5T2 1 - 112.T + 1.03e5T^{2}
53 1−211.T+1.48e5T2 1 - 211.T + 1.48e5T^{2}
59 1−439.T+2.05e5T2 1 - 439.T + 2.05e5T^{2}
61 1+227.T+2.26e5T2 1 + 227.T + 2.26e5T^{2}
67 1−723.T+3.00e5T2 1 - 723.T + 3.00e5T^{2}
71 1+1.00e3T+3.57e5T2 1 + 1.00e3T + 3.57e5T^{2}
73 1+15.8T+3.89e5T2 1 + 15.8T + 3.89e5T^{2}
79 1+972.T+4.93e5T2 1 + 972.T + 4.93e5T^{2}
83 1+155.T+5.71e5T2 1 + 155.T + 5.71e5T^{2}
89 1+517.T+7.04e5T2 1 + 517.T + 7.04e5T^{2}
97 1−1.59e3T+9.12e5T2 1 - 1.59e3T + 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

−11.12292003677402108923233324674, −10.32574826074189033302778138425, −8.589698991829085163528727467016, −7.913860149635067133960156269906, −7.36674052367949654123194237368, −5.84778493771687643946400343738, −4.89758208120329195167767685419, −4.09914408337403873325395980399, −2.55806409216120429423647926168, −0.34756446319607860798650849152, 0.34756446319607860798650849152, 2.55806409216120429423647926168, 4.09914408337403873325395980399, 4.89758208120329195167767685419, 5.84778493771687643946400343738, 7.36674052367949654123194237368, 7.913860149635067133960156269906, 8.589698991829085163528727467016, 10.32574826074189033302778138425, 11.12292003677402108923233324674

Graph of the ZZ-function along the critical line