Properties

Label 2-1840-1.1-c1-0-31
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  + 3·3-s + 5-s + 2·7-s + 6·9-s + 13-s + 3·15-s + 6·21-s − 23-s + 25-s + 9·27-s − 3·29-s − 3·31-s + 2·35-s − 8·37-s + 3·39-s + 3·41-s + 2·43-s + 6·45-s + 11·47-s − 3·49-s − 14·53-s + 8·59-s − 4·61-s + 12·63-s + 65-s + 4·67-s − 3·69-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s + 0.755·7-s + 2·9-s + 0.277·13-s + 0.774·15-s + 1.30·21-s − 0.208·23-s + 1/5·25-s + 1.73·27-s − 0.557·29-s − 0.538·31-s + 0.338·35-s − 1.31·37-s + 0.480·39-s + 0.468·41-s + 0.304·43-s + 0.894·45-s + 1.60·47-s − 3/7·49-s − 1.92·53-s + 1.04·59-s − 0.512·61-s + 1.51·63-s + 0.124·65-s + 0.488·67-s − 0.361·69-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

\(L(1)\) \(\approx\) \(3.871277106\)
\(L(\frac12)\) \(\approx\) \(3.871277106\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−8.999051054088738205599213223507, −8.642854744976674671107691765539, −7.73854228255682207536319172149, −7.26632068670665245818970027733, −6.10696554539960738301895722103, −5.02356588927784018754647193422, −4.07459093437266598889393557689, −3.25036929536985039826813471791, −2.24637242067700269920582104471, −1.50488623951800161308613033156, 1.50488623951800161308613033156, 2.24637242067700269920582104471, 3.25036929536985039826813471791, 4.07459093437266598889393557689, 5.02356588927784018754647193422, 6.10696554539960738301895722103, 7.26632068670665245818970027733, 7.73854228255682207536319172149, 8.642854744976674671107691765539, 8.999051054088738205599213223507

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