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Properties

Label 2-832-1.1-c1-0-8
Degree 22
Conductor 832832
Sign 11
Analytic cond. 6.643556.64355
Root an. cond. 2.577502.57750
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.56·3-s − 0.561·5-s + 0.561·7-s + 3.56·9-s + 2·11-s + 13-s − 1.43·15-s − 0.561·17-s + 6·19-s + 1.43·21-s − 4.68·25-s + 1.43·27-s + 8.24·29-s − 7.12·31-s + 5.12·33-s − 0.315·35-s + 9.68·37-s + 2.56·39-s + 7.12·41-s − 8.80·43-s − 2.00·45-s − 1.68·47-s − 6.68·49-s − 1.43·51-s + 4.87·53-s − 1.12·55-s + 15.3·57-s + ⋯
L(s)  = 1  + 1.47·3-s − 0.251·5-s + 0.212·7-s + 1.18·9-s + 0.603·11-s + 0.277·13-s − 0.371·15-s − 0.136·17-s + 1.37·19-s + 0.313·21-s − 0.936·25-s + 0.276·27-s + 1.53·29-s − 1.27·31-s + 0.891·33-s − 0.0533·35-s + 1.59·37-s + 0.410·39-s + 1.11·41-s − 1.34·43-s − 0.298·45-s − 0.245·47-s − 0.954·49-s − 0.201·51-s + 0.669·53-s − 0.151·55-s + 2.03·57-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: 11
Analytic conductor: 6.643556.64355
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 832, ( :1/2), 1)(2,\ 832,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6119948952.611994895
L(12)L(\frac12) ≈\approx 2.6119948952.611994895
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 1
13 1−T 1 - T
good3 1−2.56T+3T2 1 - 2.56T + 3T^{2}
5 1+0.561T+5T2 1 + 0.561T + 5T^{2}
7 1−0.561T+7T2 1 - 0.561T + 7T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
17 1+0.561T+17T2 1 + 0.561T + 17T^{2}
19 1−6T+19T2 1 - 6T + 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1−8.24T+29T2 1 - 8.24T + 29T^{2}
31 1+7.12T+31T2 1 + 7.12T + 31T^{2}
37 1−9.68T+37T2 1 - 9.68T + 37T^{2}
41 1−7.12T+41T2 1 - 7.12T + 41T^{2}
43 1+8.80T+43T2 1 + 8.80T + 43T^{2}
47 1+1.68T+47T2 1 + 1.68T + 47T^{2}
53 1−4.87T+53T2 1 - 4.87T + 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1+13.3T+61T2 1 + 13.3T + 61T^{2}
67 1−6T+67T2 1 - 6T + 67T^{2}
71 1−1.68T+71T2 1 - 1.68T + 71T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1+12T+79T2 1 + 12T + 79T^{2}
83 1+17.3T+83T2 1 + 17.3T + 83T^{2}
89 1+8.24T+89T2 1 + 8.24T + 89T^{2}
97 1+6T+97T2 1 + 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

−9.806057451218276113604895515910, −9.381712764273865955572981583725, −8.450013455410768989159259017789, −7.85722718878998315629993807171, −7.06528923023124603922603022090, −5.88208074573168203198918741286, −4.54070247726807357277017577522, −3.62176695122971953067327633570, −2.77544523177250281345606381306, −1.47481550740806929509898867304, 1.47481550740806929509898867304, 2.77544523177250281345606381306, 3.62176695122971953067327633570, 4.54070247726807357277017577522, 5.88208074573168203198918741286, 7.06528923023124603922603022090, 7.85722718878998315629993807171, 8.450013455410768989159259017789, 9.381712764273865955572981583725, 9.806057451218276113604895515910

Graph of the ZZ-function along the critical line