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Properties

Label 2-3800-1.1-c1-0-32
Degree 22
Conductor 38003800
Sign 11
Analytic cond. 30.343130.3431
Root an. cond. 5.508465.50846
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  + 0.185·3-s + 4.45·7-s − 2.96·9-s + 2.64·11-s − 1.30·13-s + 3.51·17-s − 19-s + 0.826·21-s + 6.52·23-s − 1.10·27-s − 5.20·29-s + 10.8·31-s + 0.490·33-s − 2.04·37-s − 0.241·39-s − 3.80·41-s − 4.77·43-s − 1.49·47-s + 12.8·49-s + 0.652·51-s − 0.225·53-s − 0.185·57-s − 2.86·59-s − 6.31·61-s − 13.2·63-s + 13.1·67-s + 1.21·69-s + ⋯
L(s)  = 1  + 0.107·3-s + 1.68·7-s − 0.988·9-s + 0.797·11-s − 0.360·13-s + 0.853·17-s − 0.229·19-s + 0.180·21-s + 1.36·23-s − 0.212·27-s − 0.967·29-s + 1.94·31-s + 0.0854·33-s − 0.336·37-s − 0.0386·39-s − 0.593·41-s − 0.727·43-s − 0.217·47-s + 1.83·49-s + 0.0913·51-s − 0.0309·53-s − 0.0245·57-s − 0.372·59-s − 0.808·61-s − 1.66·63-s + 1.61·67-s + 0.145·69-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 30.343130.3431
Root analytic conductor: 5.508465.50846
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3800, ( :1/2), 1)(2,\ 3800,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4842189932.484218993
L(12)L(\frac12) ≈\approx 2.4842189932.484218993
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
5 1 1
19 1+T 1 + T
good3 1−0.185T+3T2 1 - 0.185T + 3T^{2}
7 1−4.45T+7T2 1 - 4.45T + 7T^{2}
11 1−2.64T+11T2 1 - 2.64T + 11T^{2}
13 1+1.30T+13T2 1 + 1.30T + 13T^{2}
17 1−3.51T+17T2 1 - 3.51T + 17T^{2}
23 1−6.52T+23T2 1 - 6.52T + 23T^{2}
29 1+5.20T+29T2 1 + 5.20T + 29T^{2}
31 1−10.8T+31T2 1 - 10.8T + 31T^{2}
37 1+2.04T+37T2 1 + 2.04T + 37T^{2}
41 1+3.80T+41T2 1 + 3.80T + 41T^{2}
43 1+4.77T+43T2 1 + 4.77T + 43T^{2}
47 1+1.49T+47T2 1 + 1.49T + 47T^{2}
53 1+0.225T+53T2 1 + 0.225T + 53T^{2}
59 1+2.86T+59T2 1 + 2.86T + 59T^{2}
61 1+6.31T+61T2 1 + 6.31T + 61T^{2}
67 1−13.1T+67T2 1 - 13.1T + 67T^{2}
71 1−12.3T+71T2 1 - 12.3T + 71T^{2}
73 1−5.42T+73T2 1 - 5.42T + 73T^{2}
79 1−14.9T+79T2 1 - 14.9T + 79T^{2}
83 1+3.84T+83T2 1 + 3.84T + 83T^{2}
89 1−1.67T+89T2 1 - 1.67T + 89T^{2}
97 1+9.48T+97T2 1 + 9.48T + 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

−8.272624986136945238379650887192, −8.063842296408725867187989584992, −7.08185461298803165917526987803, −6.29206888048647638496356029544, −5.23525721337534450687527097188, −4.95257414709823673884497690017, −3.88847013151906623993700021614, −2.94307679812171951738207131986, −1.92945799514007907577466530205, −0.965009873455249558456152894215, 0.965009873455249558456152894215, 1.92945799514007907577466530205, 2.94307679812171951738207131986, 3.88847013151906623993700021614, 4.95257414709823673884497690017, 5.23525721337534450687527097188, 6.29206888048647638496356029544, 7.08185461298803165917526987803, 8.063842296408725867187989584992, 8.272624986136945238379650887192

Graph of the ZZ-function along the critical line