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Properties

Label 2-181944-1.1-c1-0-21
Degree 22
Conductor 181944181944
Sign 11
Analytic cond. 1452.831452.83
Root an. cond. 38.116038.1160
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 7-s + 4·11-s + 6·13-s − 2·17-s + 4·23-s − 25-s + 6·29-s − 4·31-s + 2·35-s − 2·37-s + 10·41-s + 12·43-s − 4·47-s + 49-s + 14·53-s − 8·55-s − 4·59-s + 6·61-s − 12·65-s − 4·67-s − 8·71-s − 6·73-s − 4·77-s + 12·79-s − 12·83-s + 4·85-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s + 1.20·11-s + 1.66·13-s − 0.485·17-s + 0.834·23-s − 1/5·25-s + 1.11·29-s − 0.718·31-s + 0.338·35-s − 0.328·37-s + 1.56·41-s + 1.82·43-s − 0.583·47-s + 1/7·49-s + 1.92·53-s − 1.07·55-s − 0.520·59-s + 0.768·61-s − 1.48·65-s − 0.488·67-s − 0.949·71-s − 0.702·73-s − 0.455·77-s + 1.35·79-s − 1.31·83-s + 0.433·85-s + ⋯

Functional equation

Λ(s)=(181944s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 181944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(181944s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 181944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 181944181944    =    23⋅32⋅7⋅1922^{3} \cdot 3^{2} \cdot 7 \cdot 19^{2}
Sign: 11
Analytic conductor: 1452.831452.83
Root analytic conductor: 38.116038.1160
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 181944, ( :1/2), 1)(2,\ 181944,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7169381662.716938166
L(12)L(\frac12) ≈\approx 2.7169381662.716938166
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
3 1 1
7 1+T 1 + T
19 1 1
good5 1+2T+pT2 1 + 2 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1−14T+pT2 1 - 14 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−12T+pT2 1 - 12 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+2T+pT2 1 + 2 T + p T^{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

−13.28119390117037, −12.54794512793603, −12.25331977329376, −11.65063745320705, −11.34640253202417, −10.74772473160647, −10.63381860238652, −9.705194361490352, −9.225640981328234, −8.871010873434905, −8.452469316146776, −7.943073496109164, −7.255400577870894, −6.906087162716290, −6.402967261654883, −5.852013635978511, −5.473037849279491, −4.459467862878023, −4.070687168387184, −3.882819156209606, −3.126175549858645, −2.646299801831117, −1.692642187105167, −1.070570224276452, −0.5568558116918914, 0.5568558116918914, 1.070570224276452, 1.692642187105167, 2.646299801831117, 3.126175549858645, 3.882819156209606, 4.070687168387184, 4.459467862878023, 5.473037849279491, 5.852013635978511, 6.402967261654883, 6.906087162716290, 7.255400577870894, 7.943073496109164, 8.452469316146776, 8.871010873434905, 9.225640981328234, 9.705194361490352, 10.63381860238652, 10.74772473160647, 11.34640253202417, 11.65063745320705, 12.25331977329376, 12.54794512793603, 13.28119390117037

Graph of the ZZ-function along the critical line