Properties

Label 2-1856-1.1-c3-0-12
Degree $2$
Conductor $1856$
Sign $1$
Analytic cond. $109.507$
Root an. cond. $10.4645$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.01·3-s − 0.397·5-s − 14.4·7-s − 25.9·9-s − 52.7·11-s − 60.7·13-s − 0.401·15-s + 0.0555·17-s − 100.·19-s − 14.5·21-s + 15.2·23-s − 124.·25-s − 53.5·27-s − 29·29-s + 172.·31-s − 53.3·33-s + 5.73·35-s − 305.·37-s − 61.4·39-s + 318.·41-s − 467.·43-s + 10.3·45-s + 249.·47-s − 134.·49-s + 0.0561·51-s + 201.·53-s + 20.9·55-s + ⋯
L(s)  = 1  + 0.194·3-s − 0.0355·5-s − 0.778·7-s − 0.962·9-s − 1.44·11-s − 1.29·13-s − 0.00691·15-s + 0.000792·17-s − 1.20·19-s − 0.151·21-s + 0.138·23-s − 0.998·25-s − 0.381·27-s − 0.185·29-s + 0.998·31-s − 0.281·33-s + 0.0276·35-s − 1.35·37-s − 0.252·39-s + 1.21·41-s − 1.65·43-s + 0.0341·45-s + 0.774·47-s − 0.393·49-s + 0.000154·51-s + 0.522·53-s + 0.0514·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $1$
Analytic conductor: \(109.507\)
Root analytic conductor: \(10.4645\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3672935894\)
\(L(\frac12)\) \(\approx\) \(0.3672935894\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + 29T \)
good3 \( 1 - 1.01T + 27T^{2} \)
5 \( 1 + 0.397T + 125T^{2} \)
7 \( 1 + 14.4T + 343T^{2} \)
11 \( 1 + 52.7T + 1.33e3T^{2} \)
13 \( 1 + 60.7T + 2.19e3T^{2} \)
17 \( 1 - 0.0555T + 4.91e3T^{2} \)
19 \( 1 + 100.T + 6.85e3T^{2} \)
23 \( 1 - 15.2T + 1.21e4T^{2} \)
31 \( 1 - 172.T + 2.97e4T^{2} \)
37 \( 1 + 305.T + 5.06e4T^{2} \)
41 \( 1 - 318.T + 6.89e4T^{2} \)
43 \( 1 + 467.T + 7.95e4T^{2} \)
47 \( 1 - 249.T + 1.03e5T^{2} \)
53 \( 1 - 201.T + 1.48e5T^{2} \)
59 \( 1 - 696.T + 2.05e5T^{2} \)
61 \( 1 - 796.T + 2.26e5T^{2} \)
67 \( 1 + 828.T + 3.00e5T^{2} \)
71 \( 1 - 676.T + 3.57e5T^{2} \)
73 \( 1 + 735.T + 3.89e5T^{2} \)
79 \( 1 - 149.T + 4.93e5T^{2} \)
83 \( 1 - 947.T + 5.71e5T^{2} \)
89 \( 1 + 80.8T + 7.04e5T^{2} \)
97 \( 1 + 1.03e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.760960515469599586200078467239, −8.164546539859176922532928147151, −7.37142880726088615949587180898, −6.50978874256499355602004341225, −5.60336810403663958770585940321, −4.94281166645378493514087334247, −3.78217404160499395309832210868, −2.73506504140253755132772445379, −2.24711227627512677144299460483, −0.25501605763080462599350759825, 0.25501605763080462599350759825, 2.24711227627512677144299460483, 2.73506504140253755132772445379, 3.78217404160499395309832210868, 4.94281166645378493514087334247, 5.60336810403663958770585940321, 6.50978874256499355602004341225, 7.37142880726088615949587180898, 8.164546539859176922532928147151, 8.760960515469599586200078467239

Graph of the $Z$-function along the critical line