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Properties

Label 2-7800-1.1-c1-0-4
Degree 22
Conductor 78007800
Sign 11
Analytic cond. 62.283362.2833
Root an. cond. 7.891977.89197
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 + 9-s − 4.64·11-s − 13-s − 4.24·17-s − 6.24·19-s − 2.24·23-s − 27-s − 9.21·29-s + 9.28·31-s + 4.64·33-s − 7.28·37-s + 39-s − 5.67·41-s + 4.24·43-s − 2.88·47-s − 7·49-s + 4.24·51-s + 9.21·53-s + 6.24·57-s + 5.92·59-s + 0.969·61-s + 1.93·67-s + 2.24·69-s + 5.60·71-s + 12.5·73-s + 12.2·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.333·9-s − 1.39·11-s − 0.277·13-s − 1.03·17-s − 1.43·19-s − 0.469·23-s − 0.192·27-s − 1.71·29-s + 1.66·31-s + 0.807·33-s − 1.19·37-s + 0.160·39-s − 0.885·41-s + 0.648·43-s − 0.421·47-s − 49-s + 0.595·51-s + 1.26·53-s + 0.827·57-s + 0.770·59-s + 0.124·61-s + 0.236·67-s + 0.270·69-s + 0.665·71-s + 1.47·73-s + 1.37·79-s + ⋯

Functional equation

Λ(s)=(7800s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7800s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 78007800    =    23⋅3⋅52⋅132^{3} \cdot 3 \cdot 5^{2} \cdot 13
Sign: 11
Analytic conductor: 62.283362.2833
Root analytic conductor: 7.891977.89197
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7800, ( :1/2), 1)(2,\ 7800,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.61900380810.6190038081
L(12)L(\frac12) ≈\approx 0.61900380810.6190038081
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 1
13 1+T 1 + T
good7 1+7T2 1 + 7T^{2}
11 1+4.64T+11T2 1 + 4.64T + 11T^{2}
17 1+4.24T+17T2 1 + 4.24T + 17T^{2}
19 1+6.24T+19T2 1 + 6.24T + 19T^{2}
23 1+2.24T+23T2 1 + 2.24T + 23T^{2}
29 1+9.21T+29T2 1 + 9.21T + 29T^{2}
31 1−9.28T+31T2 1 - 9.28T + 31T^{2}
37 1+7.28T+37T2 1 + 7.28T + 37T^{2}
41 1+5.67T+41T2 1 + 5.67T + 41T^{2}
43 1−4.24T+43T2 1 - 4.24T + 43T^{2}
47 1+2.88T+47T2 1 + 2.88T + 47T^{2}
53 1−9.21T+53T2 1 - 9.21T + 53T^{2}
59 1−5.92T+59T2 1 - 5.92T + 59T^{2}
61 1−0.969T+61T2 1 - 0.969T + 61T^{2}
67 1−1.93T+67T2 1 - 1.93T + 67T^{2}
71 1−5.60T+71T2 1 - 5.60T + 71T^{2}
73 1−12.5T+73T2 1 - 12.5T + 73T^{2}
79 1−12.2T+79T2 1 - 12.2T + 79T^{2}
83 1−3.67T+83T2 1 - 3.67T + 83T^{2}
89 1+9.67T+89T2 1 + 9.67T + 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

−8.006969271006219103153679067393, −6.96538411088603795364213743129, −6.57616106872550525270366432391, −5.68698586867143787130507874959, −5.10106057390892458554905689707, −4.43099735751405480050994468139, −3.63362864948348296696776105558, −2.47150415270460808491945962002, −1.93893175193214366695245804105, −0.37878694050747900260301198063, 0.37878694050747900260301198063, 1.93893175193214366695245804105, 2.47150415270460808491945962002, 3.63362864948348296696776105558, 4.43099735751405480050994468139, 5.10106057390892458554905689707, 5.68698586867143787130507874959, 6.57616106872550525270366432391, 6.96538411088603795364213743129, 8.006969271006219103153679067393

Graph of the ZZ-function along the critical line