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Properties

Label 2-936-1.1-c3-0-17
Degree 22
Conductor 936936
Sign 11
Analytic cond. 55.225755.2257
Root an. cond. 7.431407.43140
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  + 7.29·5-s + 5.87·7-s + 51.1·11-s − 13·13-s + 73.7·17-s + 59.9·19-s − 69.8·23-s − 71.8·25-s + 294.·29-s − 334.·31-s + 42.8·35-s + 261.·37-s − 222.·41-s + 79.2·43-s + 584.·47-s − 308.·49-s − 465.·53-s + 373.·55-s + 530.·59-s + 548.·61-s − 94.7·65-s − 384.·67-s + 307.·71-s − 844.·73-s + 300.·77-s + 30.1·79-s − 19.5·83-s + ⋯
L(s)  = 1  + 0.652·5-s + 0.317·7-s + 1.40·11-s − 0.277·13-s + 1.05·17-s + 0.723·19-s − 0.633·23-s − 0.574·25-s + 1.88·29-s − 1.93·31-s + 0.206·35-s + 1.16·37-s − 0.848·41-s + 0.281·43-s + 1.81·47-s − 0.899·49-s − 1.20·53-s + 0.914·55-s + 1.16·59-s + 1.15·61-s − 0.180·65-s − 0.701·67-s + 0.513·71-s − 1.35·73-s + 0.444·77-s + 0.0429·79-s − 0.0258·83-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: 11
Analytic conductor: 55.225755.2257
Root analytic conductor: 7.431407.43140
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 936, ( :3/2), 1)(2,\ 936,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.8608780752.860878075
L(12)L(\frac12) ≈\approx 2.8608780752.860878075
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
3 1 1
13 1+13T 1 + 13T
good5 1−7.29T+125T2 1 - 7.29T + 125T^{2}
7 1−5.87T+343T2 1 - 5.87T + 343T^{2}
11 1−51.1T+1.33e3T2 1 - 51.1T + 1.33e3T^{2}
17 1−73.7T+4.91e3T2 1 - 73.7T + 4.91e3T^{2}
19 1−59.9T+6.85e3T2 1 - 59.9T + 6.85e3T^{2}
23 1+69.8T+1.21e4T2 1 + 69.8T + 1.21e4T^{2}
29 1−294.T+2.43e4T2 1 - 294.T + 2.43e4T^{2}
31 1+334.T+2.97e4T2 1 + 334.T + 2.97e4T^{2}
37 1−261.T+5.06e4T2 1 - 261.T + 5.06e4T^{2}
41 1+222.T+6.89e4T2 1 + 222.T + 6.89e4T^{2}
43 1−79.2T+7.95e4T2 1 - 79.2T + 7.95e4T^{2}
47 1−584.T+1.03e5T2 1 - 584.T + 1.03e5T^{2}
53 1+465.T+1.48e5T2 1 + 465.T + 1.48e5T^{2}
59 1−530.T+2.05e5T2 1 - 530.T + 2.05e5T^{2}
61 1−548.T+2.26e5T2 1 - 548.T + 2.26e5T^{2}
67 1+384.T+3.00e5T2 1 + 384.T + 3.00e5T^{2}
71 1−307.T+3.57e5T2 1 - 307.T + 3.57e5T^{2}
73 1+844.T+3.89e5T2 1 + 844.T + 3.89e5T^{2}
79 1−30.1T+4.93e5T2 1 - 30.1T + 4.93e5T^{2}
83 1+19.5T+5.71e5T2 1 + 19.5T + 5.71e5T^{2}
89 1−513.T+7.04e5T2 1 - 513.T + 7.04e5T^{2}
97 1−787.T+9.12e5T2 1 - 787.T + 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

−9.682769275316558451439543479409, −8.989548486388880517970525139888, −7.986602093722320726290349830754, −7.11642283516373095862941225541, −6.15425401009738511870859127689, −5.43878472926334727798313626109, −4.31517351953994171765573131581, −3.30871327573630775011803171414, −1.96978322746292201857488326676, −0.980845958631757754395181003044, 0.980845958631757754395181003044, 1.96978322746292201857488326676, 3.30871327573630775011803171414, 4.31517351953994171765573131581, 5.43878472926334727798313626109, 6.15425401009738511870859127689, 7.11642283516373095862941225541, 7.986602093722320726290349830754, 8.989548486388880517970525139888, 9.682769275316558451439543479409

Graph of the ZZ-function along the critical line