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Properties

Label 2-3800-1.1-c1-0-73
Degree 22
Conductor 38003800
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.59·3-s − 1.66·7-s − 0.463·9-s + 5.56·11-s − 6.31·13-s − 4.12·17-s + 19-s − 2.64·21-s + 1.82·23-s − 5.51·27-s − 4.08·29-s + 6.61·31-s + 8.86·33-s − 9.66·37-s − 10.0·39-s − 4.61·41-s + 3.75·43-s + 3.85·47-s − 4.23·49-s − 6.56·51-s − 5.24·53-s + 1.59·57-s − 11.5·59-s + 8.02·61-s + 0.770·63-s − 0.155·67-s + 2.90·69-s + ⋯
L(s)  = 1  + 0.919·3-s − 0.628·7-s − 0.154·9-s + 1.67·11-s − 1.75·13-s − 0.999·17-s + 0.229·19-s − 0.577·21-s + 0.380·23-s − 1.06·27-s − 0.759·29-s + 1.18·31-s + 1.54·33-s − 1.58·37-s − 1.61·39-s − 0.720·41-s + 0.572·43-s + 0.562·47-s − 0.604·49-s − 0.919·51-s − 0.720·53-s + 0.210·57-s − 1.50·59-s + 1.02·61-s + 0.0970·63-s − 0.0189·67-s + 0.349·69-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: −1-1
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: 11
Selberg data: (2, 3800, ( :1/2), −1)(2,\ 3800,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1−1.59T+3T2 1 - 1.59T + 3T^{2}
7 1+1.66T+7T2 1 + 1.66T + 7T^{2}
11 1−5.56T+11T2 1 - 5.56T + 11T^{2}
13 1+6.31T+13T2 1 + 6.31T + 13T^{2}
17 1+4.12T+17T2 1 + 4.12T + 17T^{2}
23 1−1.82T+23T2 1 - 1.82T + 23T^{2}
29 1+4.08T+29T2 1 + 4.08T + 29T^{2}
31 1−6.61T+31T2 1 - 6.61T + 31T^{2}
37 1+9.66T+37T2 1 + 9.66T + 37T^{2}
41 1+4.61T+41T2 1 + 4.61T + 41T^{2}
43 1−3.75T+43T2 1 - 3.75T + 43T^{2}
47 1−3.85T+47T2 1 - 3.85T + 47T^{2}
53 1+5.24T+53T2 1 + 5.24T + 53T^{2}
59 1+11.5T+59T2 1 + 11.5T + 59T^{2}
61 1−8.02T+61T2 1 - 8.02T + 61T^{2}
67 1+0.155T+67T2 1 + 0.155T + 67T^{2}
71 1+12.1T+71T2 1 + 12.1T + 71T^{2}
73 1+0.795T+73T2 1 + 0.795T + 73T^{2}
79 1+7.18T+79T2 1 + 7.18T + 79T^{2}
83 1+5.88T+83T2 1 + 5.88T + 83T^{2}
89 1−18.5T+89T2 1 - 18.5T + 89T^{2}
97 1+2.77T+97T2 1 + 2.77T + 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

−8.262434657392541449887795408428, −7.30055714268107584107407746459, −6.83366008486754876346150341082, −6.04803533508830546684026510723, −4.97222295852680425058959755904, −4.16564523352090037701538168768, −3.32255847924512262847848311483, −2.60743172684876293927640633048, −1.66661355496970272230647656251, 0, 1.66661355496970272230647656251, 2.60743172684876293927640633048, 3.32255847924512262847848311483, 4.16564523352090037701538168768, 4.97222295852680425058959755904, 6.04803533508830546684026510723, 6.83366008486754876346150341082, 7.30055714268107584107407746459, 8.262434657392541449887795408428

Graph of the ZZ-function along the critical line