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Properties

Label 2-7800-1.1-c1-0-35
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 + 4.83·7-s + 9-s + 2.25·11-s + 13-s − 6.32·17-s − 2.58·19-s − 4.83·21-s − 4.83·23-s − 27-s + 9.09·29-s − 0.510·31-s − 2.25·33-s − 4.25·37-s − 39-s + 0.255·41-s − 9.16·43-s + 4.51·47-s + 16.4·49-s + 6.32·51-s + 9.93·53-s + 2.58·57-s + 14.1·59-s + 9.93·61-s + 4.83·63-s + 13.1·67-s + 4.83·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.82·7-s + 0.333·9-s + 0.679·11-s + 0.277·13-s − 1.53·17-s − 0.592·19-s − 1.05·21-s − 1.00·23-s − 0.192·27-s + 1.68·29-s − 0.0916·31-s − 0.392·33-s − 0.699·37-s − 0.160·39-s + 0.0398·41-s − 1.39·43-s + 0.657·47-s + 2.34·49-s + 0.886·51-s + 1.36·53-s + 0.342·57-s + 1.84·59-s + 1.27·61-s + 0.609·63-s + 1.60·67-s + 0.582·69-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 2.1265501592.126550159
L(12)L(\frac12) ≈\approx 2.1265501592.126550159
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 1−4.83T+7T2 1 - 4.83T + 7T^{2}
11 1−2.25T+11T2 1 - 2.25T + 11T^{2}
17 1+6.32T+17T2 1 + 6.32T + 17T^{2}
19 1+2.58T+19T2 1 + 2.58T + 19T^{2}
23 1+4.83T+23T2 1 + 4.83T + 23T^{2}
29 1−9.09T+29T2 1 - 9.09T + 29T^{2}
31 1+0.510T+31T2 1 + 0.510T + 31T^{2}
37 1+4.25T+37T2 1 + 4.25T + 37T^{2}
41 1−0.255T+41T2 1 - 0.255T + 41T^{2}
43 1+9.16T+43T2 1 + 9.16T + 43T^{2}
47 1−4.51T+47T2 1 - 4.51T + 47T^{2}
53 1−9.93T+53T2 1 - 9.93T + 53T^{2}
59 1−14.1T+59T2 1 - 14.1T + 59T^{2}
61 1−9.93T+61T2 1 - 9.93T + 61T^{2}
67 1−13.1T+67T2 1 - 13.1T + 67T^{2}
71 1+5.74T+71T2 1 + 5.74T + 71T^{2}
73 1+2.90T+73T2 1 + 2.90T + 73T^{2}
79 1−5.74T+79T2 1 - 5.74T + 79T^{2}
83 1+10.1T+83T2 1 + 10.1T + 83T^{2}
89 1−3.74T+89T2 1 - 3.74T + 89T^{2}
97 1−12.5T+97T2 1 - 12.5T + 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

−8.011624286498535626592586537675, −6.93041136517787506798242368227, −6.63871240153221342085490185068, −5.66678247383174095845090383777, −5.01919880369671783130541656517, −4.34240483211921496672096916757, −3.89840924713107628819073531767, −2.35206918202470307670813960654, −1.77866166453167689708972984679, −0.77465680041494378385884869220, 0.77465680041494378385884869220, 1.77866166453167689708972984679, 2.35206918202470307670813960654, 3.89840924713107628819073531767, 4.34240483211921496672096916757, 5.01919880369671783130541656517, 5.66678247383174095845090383777, 6.63871240153221342085490185068, 6.93041136517787506798242368227, 8.011624286498535626592586537675

Graph of the ZZ-function along the critical line