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Properties

Label 2-3800-1.1-c1-0-24
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  + 0.471·3-s + 0.567·7-s − 2.77·9-s + 4.37·11-s − 0.165·13-s − 7.94·17-s + 19-s + 0.267·21-s + 3.87·23-s − 2.72·27-s + 3.53·29-s + 3.20·31-s + 2.06·33-s + 10.1·37-s − 0.0779·39-s − 5.97·41-s + 12.0·43-s − 5.46·47-s − 6.67·49-s − 3.74·51-s − 2.00·53-s + 0.471·57-s + 8.32·59-s + 11.8·61-s − 1.57·63-s + 8.79·67-s + 1.82·69-s + ⋯
L(s)  = 1  + 0.271·3-s + 0.214·7-s − 0.926·9-s + 1.32·11-s − 0.0458·13-s − 1.92·17-s + 0.229·19-s + 0.0583·21-s + 0.807·23-s − 0.523·27-s + 0.655·29-s + 0.575·31-s + 0.358·33-s + 1.67·37-s − 0.0124·39-s − 0.932·41-s + 1.84·43-s − 0.796·47-s − 0.954·49-s − 0.524·51-s − 0.275·53-s + 0.0623·57-s + 1.08·59-s + 1.51·61-s − 0.198·63-s + 1.07·67-s + 0.219·69-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 2.0331541342.033154134
L(12)L(\frac12) ≈\approx 2.0331541342.033154134
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 1−0.471T+3T2 1 - 0.471T + 3T^{2}
7 1−0.567T+7T2 1 - 0.567T + 7T^{2}
11 1−4.37T+11T2 1 - 4.37T + 11T^{2}
13 1+0.165T+13T2 1 + 0.165T + 13T^{2}
17 1+7.94T+17T2 1 + 7.94T + 17T^{2}
23 1−3.87T+23T2 1 - 3.87T + 23T^{2}
29 1−3.53T+29T2 1 - 3.53T + 29T^{2}
31 1−3.20T+31T2 1 - 3.20T + 31T^{2}
37 1−10.1T+37T2 1 - 10.1T + 37T^{2}
41 1+5.97T+41T2 1 + 5.97T + 41T^{2}
43 1−12.0T+43T2 1 - 12.0T + 43T^{2}
47 1+5.46T+47T2 1 + 5.46T + 47T^{2}
53 1+2.00T+53T2 1 + 2.00T + 53T^{2}
59 1−8.32T+59T2 1 - 8.32T + 59T^{2}
61 1−11.8T+61T2 1 - 11.8T + 61T^{2}
67 1−8.79T+67T2 1 - 8.79T + 67T^{2}
71 1−0.720T+71T2 1 - 0.720T + 71T^{2}
73 1−4.54T+73T2 1 - 4.54T + 73T^{2}
79 1−11.7T+79T2 1 - 11.7T + 79T^{2}
83 1+6.72T+83T2 1 + 6.72T + 83T^{2}
89 1−8.11T+89T2 1 - 8.11T + 89T^{2}
97 1+13.6T+97T2 1 + 13.6T + 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

−8.500193341886418798288013217090, −7.984820849934032354252099659200, −6.70582164544817996685952791504, −6.58743311623019396398719591619, −5.50809319115833067201241661972, −4.60314274779459705053605964331, −3.93283401751500165448886812878, −2.89417324206712532907744656805, −2.11319549206465084375759513915, −0.818379065893886754149673685414, 0.818379065893886754149673685414, 2.11319549206465084375759513915, 2.89417324206712532907744656805, 3.93283401751500165448886812878, 4.60314274779459705053605964331, 5.50809319115833067201241661972, 6.58743311623019396398719591619, 6.70582164544817996685952791504, 7.984820849934032354252099659200, 8.500193341886418798288013217090

Graph of the ZZ-function along the critical line