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Properties

Label 2-38-1.1-c5-0-3
Degree 22
Conductor 3838
Sign 11
Analytic cond. 6.094586.09458
Root an. cond. 2.468722.46872
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 14.7·3-s + 16·4-s − 19.0·5-s + 59.1·6-s + 212.·7-s + 64·8-s − 23.9·9-s − 76.3·10-s − 662.·11-s + 236.·12-s + 1.11e3·13-s + 849.·14-s − 282.·15-s + 256·16-s − 522.·17-s − 95.9·18-s − 361·19-s − 305.·20-s + 3.14e3·21-s − 2.65e3·22-s − 3.21e3·23-s + 947.·24-s − 2.76e3·25-s + 4.44e3·26-s − 3.95e3·27-s + 3.39e3·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.949·3-s + 0.5·4-s − 0.341·5-s + 0.671·6-s + 1.63·7-s + 0.353·8-s − 0.0987·9-s − 0.241·10-s − 1.65·11-s + 0.474·12-s + 1.82·13-s + 1.15·14-s − 0.324·15-s + 0.250·16-s − 0.438·17-s − 0.0698·18-s − 0.229·19-s − 0.170·20-s + 1.55·21-s − 1.16·22-s − 1.26·23-s + 0.335·24-s − 0.883·25-s + 1.29·26-s − 1.04·27-s + 0.818·28-s + ⋯

Functional equation

Λ(s)=(38s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(38s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3838    =    2⋅192 \cdot 19
Sign: 11
Analytic conductor: 6.094586.09458
Root analytic conductor: 2.468722.46872
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 38, ( :5/2), 1)(2,\ 38,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.0922122033.092212203
L(12)L(\frac12) ≈\approx 3.0922122033.092212203
L(72)L(\frac{7}{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 1−4T 1 - 4T
19 1+361T 1 + 361T
good3 1−14.7T+243T2 1 - 14.7T + 243T^{2}
5 1+19.0T+3.12e3T2 1 + 19.0T + 3.12e3T^{2}
7 1−212.T+1.68e4T2 1 - 212.T + 1.68e4T^{2}
11 1+662.T+1.61e5T2 1 + 662.T + 1.61e5T^{2}
13 1−1.11e3T+3.71e5T2 1 - 1.11e3T + 3.71e5T^{2}
17 1+522.T+1.41e6T2 1 + 522.T + 1.41e6T^{2}
23 1+3.21e3T+6.43e6T2 1 + 3.21e3T + 6.43e6T^{2}
29 1+3.72e3T+2.05e7T2 1 + 3.72e3T + 2.05e7T^{2}
31 1−4.83e3T+2.86e7T2 1 - 4.83e3T + 2.86e7T^{2}
37 1+7.15e3T+6.93e7T2 1 + 7.15e3T + 6.93e7T^{2}
41 1−1.08e4T+1.15e8T2 1 - 1.08e4T + 1.15e8T^{2}
43 1+1.04e4T+1.47e8T2 1 + 1.04e4T + 1.47e8T^{2}
47 1−1.19e4T+2.29e8T2 1 - 1.19e4T + 2.29e8T^{2}
53 1−2.49e4T+4.18e8T2 1 - 2.49e4T + 4.18e8T^{2}
59 1−2.32e4T+7.14e8T2 1 - 2.32e4T + 7.14e8T^{2}
61 1+1.51e4T+8.44e8T2 1 + 1.51e4T + 8.44e8T^{2}
67 1−5.92e4T+1.35e9T2 1 - 5.92e4T + 1.35e9T^{2}
71 1−1.46e4T+1.80e9T2 1 - 1.46e4T + 1.80e9T^{2}
73 1+1.89e4T+2.07e9T2 1 + 1.89e4T + 2.07e9T^{2}
79 1+3.62e3T+3.07e9T2 1 + 3.62e3T + 3.07e9T^{2}
83 1−2.26e4T+3.93e9T2 1 - 2.26e4T + 3.93e9T^{2}
89 1−6.56e4T+5.58e9T2 1 - 6.56e4T + 5.58e9T^{2}
97 1+5.94e4T+8.58e9T2 1 + 5.94e4T + 8.58e9T^{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

−15.17548649197583291338304815093, −13.99032078772933234410068781028, −13.35489327542001597598119056591, −11.61672762616855153689955877291, −10.69122895546949054067932679401, −8.421352315317977076240415353734, −7.87353626410628875190498803639, −5.57548770143407297709892494447, −3.94775800621565179734210641580, −2.15048141749809603558933299848, 2.15048141749809603558933299848, 3.94775800621565179734210641580, 5.57548770143407297709892494447, 7.87353626410628875190498803639, 8.421352315317977076240415353734, 10.69122895546949054067932679401, 11.61672762616855153689955877291, 13.35489327542001597598119056591, 13.99032078772933234410068781028, 15.17548649197583291338304815093

Graph of the ZZ-function along the critical line