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Properties

Label 2-3800-1.1-c1-0-20
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  − 2.79·3-s − 0.127·7-s + 4.83·9-s + 5.21·11-s + 0.515·13-s − 3.58·17-s + 19-s + 0.357·21-s + 6.50·23-s − 5.14·27-s − 3.05·29-s + 5.44·31-s − 14.5·33-s + 4.99·37-s − 1.44·39-s + 11.4·41-s + 2.06·43-s − 11.9·47-s − 6.98·49-s + 10.0·51-s − 2.25·53-s − 2.79·57-s + 1.89·59-s − 5.83·61-s − 0.616·63-s − 0.432·67-s − 18.2·69-s + ⋯
L(s)  = 1  − 1.61·3-s − 0.0482·7-s + 1.61·9-s + 1.57·11-s + 0.142·13-s − 0.868·17-s + 0.229·19-s + 0.0779·21-s + 1.35·23-s − 0.989·27-s − 0.567·29-s + 0.977·31-s − 2.54·33-s + 0.821·37-s − 0.231·39-s + 1.78·41-s + 0.315·43-s − 1.73·47-s − 0.997·49-s + 1.40·51-s − 0.310·53-s − 0.370·57-s + 0.246·59-s − 0.746·61-s − 0.0777·63-s − 0.0528·67-s − 2.19·69-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 1.1302776001.130277600
L(12)L(\frac12) ≈\approx 1.1302776001.130277600
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+2.79T+3T2 1 + 2.79T + 3T^{2}
7 1+0.127T+7T2 1 + 0.127T + 7T^{2}
11 1−5.21T+11T2 1 - 5.21T + 11T^{2}
13 1−0.515T+13T2 1 - 0.515T + 13T^{2}
17 1+3.58T+17T2 1 + 3.58T + 17T^{2}
23 1−6.50T+23T2 1 - 6.50T + 23T^{2}
29 1+3.05T+29T2 1 + 3.05T + 29T^{2}
31 1−5.44T+31T2 1 - 5.44T + 31T^{2}
37 1−4.99T+37T2 1 - 4.99T + 37T^{2}
41 1−11.4T+41T2 1 - 11.4T + 41T^{2}
43 1−2.06T+43T2 1 - 2.06T + 43T^{2}
47 1+11.9T+47T2 1 + 11.9T + 47T^{2}
53 1+2.25T+53T2 1 + 2.25T + 53T^{2}
59 1−1.89T+59T2 1 - 1.89T + 59T^{2}
61 1+5.83T+61T2 1 + 5.83T + 61T^{2}
67 1+0.432T+67T2 1 + 0.432T + 67T^{2}
71 1−4.77T+71T2 1 - 4.77T + 71T^{2}
73 1+9.98T+73T2 1 + 9.98T + 73T^{2}
79 1+3.28T+79T2 1 + 3.28T + 79T^{2}
83 1+0.496T+83T2 1 + 0.496T + 83T^{2}
89 1+12.4T+89T2 1 + 12.4T + 89T^{2}
97 1+7.00T+97T2 1 + 7.00T + 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

−8.608745651008628304098962327464, −7.49163907542998302063450450876, −6.72102504178230645985307422390, −6.33086137809520624732184850792, −5.64811165387960118728315524922, −4.65869862139211995941945149502, −4.26604169880664857795514823217, −3.06957168552183725983460075786, −1.58445090308576382903331718354, −0.71925271457209539483102620124, 0.71925271457209539483102620124, 1.58445090308576382903331718354, 3.06957168552183725983460075786, 4.26604169880664857795514823217, 4.65869862139211995941945149502, 5.64811165387960118728315524922, 6.33086137809520624732184850792, 6.72102504178230645985307422390, 7.49163907542998302063450450876, 8.608745651008628304098962327464

Graph of the ZZ-function along the critical line