Properties

Label 2-1840-1.1-c1-0-19
Degree $2$
Conductor $1840$
Sign $1$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 2·7-s − 2·9-s + 4·11-s + 13-s + 15-s + 4·19-s + 2·21-s + 23-s + 25-s − 5·27-s − 7·29-s + 7·31-s + 4·33-s + 2·35-s − 4·37-s + 39-s + 3·41-s − 6·43-s − 2·45-s + 13·47-s − 3·49-s + 10·53-s + 4·55-s + 4·57-s + 8·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.755·7-s − 2/3·9-s + 1.20·11-s + 0.277·13-s + 0.258·15-s + 0.917·19-s + 0.436·21-s + 0.208·23-s + 1/5·25-s − 0.962·27-s − 1.29·29-s + 1.25·31-s + 0.696·33-s + 0.338·35-s − 0.657·37-s + 0.160·39-s + 0.468·41-s − 0.914·43-s − 0.298·45-s + 1.89·47-s − 3/7·49-s + 1.37·53-s + 0.539·55-s + 0.529·57-s + 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1840} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.651321444\)
\(L(\frac12)\) \(\approx\) \(2.651321444\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
23 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−9.059985080884665135816954636616, −8.654990800185878310775361395649, −7.76306365644341418753093581036, −6.96187056242752254336518124104, −5.96476036111847992968770925817, −5.29092737891941653946525021688, −4.17812676432896289830736050198, −3.30948108344784044658086996349, −2.23725050288066304033836255143, −1.19054713736220218140741901981, 1.19054713736220218140741901981, 2.23725050288066304033836255143, 3.30948108344784044658086996349, 4.17812676432896289830736050198, 5.29092737891941653946525021688, 5.96476036111847992968770925817, 6.96187056242752254336518124104, 7.76306365644341418753093581036, 8.654990800185878310775361395649, 9.059985080884665135816954636616

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