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Properties

Label 2-3872-1.1-c1-0-66
Degree 22
Conductor 38723872
Sign 11
Analytic cond. 30.918030.9180
Root an. cond. 5.560405.56040
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  + 3·3-s + 5-s + 6·9-s + 6·13-s + 3·15-s + 4·17-s − 6·19-s + 3·23-s − 4·25-s + 9·27-s + 4·29-s − 9·31-s + 7·37-s + 18·39-s + 2·41-s − 6·43-s + 6·45-s + 12·47-s − 7·49-s + 12·51-s + 2·53-s − 18·57-s + 9·59-s − 8·61-s + 6·65-s − 15·67-s + 9·69-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s + 2·9-s + 1.66·13-s + 0.774·15-s + 0.970·17-s − 1.37·19-s + 0.625·23-s − 4/5·25-s + 1.73·27-s + 0.742·29-s − 1.61·31-s + 1.15·37-s + 2.88·39-s + 0.312·41-s − 0.914·43-s + 0.894·45-s + 1.75·47-s − 49-s + 1.68·51-s + 0.274·53-s − 2.38·57-s + 1.17·59-s − 1.02·61-s + 0.744·65-s − 1.83·67-s + 1.08·69-s + ⋯

Functional equation

Λ(s)=(3872s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3872s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38723872    =    25⋅1122^{5} \cdot 11^{2}
Sign: 11
Analytic conductor: 30.918030.9180
Root analytic conductor: 5.560405.56040
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3872, ( :1/2), 1)(2,\ 3872,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.4566460774.456646077
L(12)L(\frac12) ≈\approx 4.4566460774.456646077
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
11 1 1
good3 1−pT+pT2 1 - p T + p T^{2}
5 1−T+pT2 1 - T + p T^{2}
7 1+pT2 1 + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1−4T+pT2 1 - 4 T + p T^{2}
31 1+9T+pT2 1 + 9 T + p T^{2}
37 1−7T+pT2 1 - 7 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−9T+pT2 1 - 9 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1+15T+pT2 1 + 15 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−6T+pT2 1 - 6 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+5T+pT2 1 + 5 T + p T^{2}
97 1+3T+pT2 1 + 3 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

−8.463805274480032902771890485211, −8.006102220895286729861212502768, −7.20716272694663712940001591257, −6.32385710458151688343766354254, −5.60610239516517308689450757263, −4.32368875597613387288473485169, −3.72477499010942593504003294230, −2.99122228122758599463705875125, −2.07332820410151415501700977303, −1.27300481642330281094266289768, 1.27300481642330281094266289768, 2.07332820410151415501700977303, 2.99122228122758599463705875125, 3.72477499010942593504003294230, 4.32368875597613387288473485169, 5.60610239516517308689450757263, 6.32385710458151688343766354254, 7.20716272694663712940001591257, 8.006102220895286729861212502768, 8.463805274480032902771890485211

Graph of the ZZ-function along the critical line