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Properties

Label 2-3840-1.1-c1-0-14
Degree 22
Conductor 38403840
Sign 11
Analytic cond. 30.662530.6625
Root an. cond. 5.537375.53737
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  − 3-s + 5-s + 3.62·7-s + 9-s − 6.20·11-s − 0.578·13-s − 15-s + 1.42·17-s − 5.62·19-s − 3.62·21-s + 5.62·23-s + 25-s − 27-s + 2·29-s − 2.57·31-s + 6.20·33-s + 3.62·35-s + 7.83·37-s + 0.578·39-s − 5.25·41-s + 7.25·43-s + 45-s + 6.78·47-s + 6.15·49-s − 1.42·51-s + 2·53-s − 6.20·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.37·7-s + 0.333·9-s − 1.87·11-s − 0.160·13-s − 0.258·15-s + 0.344·17-s − 1.29·19-s − 0.791·21-s + 1.17·23-s + 0.200·25-s − 0.192·27-s + 0.371·29-s − 0.463·31-s + 1.08·33-s + 0.613·35-s + 1.28·37-s + 0.0926·39-s − 0.820·41-s + 1.10·43-s + 0.149·45-s + 0.989·47-s + 0.879·49-s − 0.199·51-s + 0.274·53-s − 0.836·55-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: 11
Analytic conductor: 30.662530.6625
Root analytic conductor: 5.537375.53737
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3840, ( :1/2), 1)(2,\ 3840,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7193876701.719387670
L(12)L(\frac12) ≈\approx 1.7193876701.719387670
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
3 1+T 1 + T
5 1−T 1 - T
good7 1−3.62T+7T2 1 - 3.62T + 7T^{2}
11 1+6.20T+11T2 1 + 6.20T + 11T^{2}
13 1+0.578T+13T2 1 + 0.578T + 13T^{2}
17 1−1.42T+17T2 1 - 1.42T + 17T^{2}
19 1+5.62T+19T2 1 + 5.62T + 19T^{2}
23 1−5.62T+23T2 1 - 5.62T + 23T^{2}
29 1−2T+29T2 1 - 2T + 29T^{2}
31 1+2.57T+31T2 1 + 2.57T + 31T^{2}
37 1−7.83T+37T2 1 - 7.83T + 37T^{2}
41 1+5.25T+41T2 1 + 5.25T + 41T^{2}
43 1−7.25T+43T2 1 - 7.25T + 43T^{2}
47 1−6.78T+47T2 1 - 6.78T + 47T^{2}
53 1−2T+53T2 1 - 2T + 53T^{2}
59 1−2.20T+59T2 1 - 2.20T + 59T^{2}
61 1−12.4T+61T2 1 - 12.4T + 61T^{2}
67 1−4T+67T2 1 - 4T + 67T^{2}
71 1+8.41T+71T2 1 + 8.41T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1−5.42T+79T2 1 - 5.42T + 79T^{2}
83 1−3.25T+83T2 1 - 3.25T + 83T^{2}
89 1−13.2T+89T2 1 - 13.2T + 89T^{2}
97 1−4.84T+97T2 1 - 4.84T + 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

−8.337702114404272222902412693485, −7.79749667720417596638601843407, −7.10357676469451190764930613242, −6.12692934387805100168835692762, −5.27142135551028810300695850719, −5.01402190397445052088750995500, −4.12381235219561182874213259525, −2.69659815563183594262504280023, −2.03621277886311350921908334777, −0.78155673211614980318945705187, 0.78155673211614980318945705187, 2.03621277886311350921908334777, 2.69659815563183594262504280023, 4.12381235219561182874213259525, 5.01402190397445052088750995500, 5.27142135551028810300695850719, 6.12692934387805100168835692762, 7.10357676469451190764930613242, 7.79749667720417596638601843407, 8.337702114404272222902412693485

Graph of the ZZ-function along the critical line