Properties

Label 2-1840-1.1-c1-0-4
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  − 5-s − 7-s − 3·9-s − 2·11-s − 2·13-s + 3·17-s + 2·19-s − 23-s + 25-s + 7·29-s + 5·31-s + 35-s + 11·37-s + 41-s + 3·45-s − 6·49-s + 11·53-s + 2·55-s + 13·59-s − 8·61-s + 3·63-s + 2·65-s − 5·67-s − 5·71-s + 6·73-s + 2·77-s + 12·79-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 9-s − 0.603·11-s − 0.554·13-s + 0.727·17-s + 0.458·19-s − 0.208·23-s + 1/5·25-s + 1.29·29-s + 0.898·31-s + 0.169·35-s + 1.80·37-s + 0.156·41-s + 0.447·45-s − 6/7·49-s + 1.51·53-s + 0.269·55-s + 1.69·59-s − 1.02·61-s + 0.377·63-s + 0.248·65-s − 0.610·67-s − 0.593·71-s + 0.702·73-s + 0.227·77-s + 1.35·79-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

\(L(1)\) \(\approx\) \(1.190913599\)
\(L(\frac12)\) \(\approx\) \(1.190913599\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−9.287436056802008861665654849050, −8.261450062978304340130490972772, −7.88313741440253118524665483256, −6.89664930841244341712826498598, −6.00165457583941898386124574131, −5.23992404060255109399587286374, −4.32075270011969972226267491278, −3.14448167474562522721276000501, −2.55324698595565232433692561243, −0.72059586748507280140651813487, 0.72059586748507280140651813487, 2.55324698595565232433692561243, 3.14448167474562522721276000501, 4.32075270011969972226267491278, 5.23992404060255109399587286374, 6.00165457583941898386124574131, 6.89664930841244341712826498598, 7.88313741440253118524665483256, 8.261450062978304340130490972772, 9.287436056802008861665654849050

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