Properties

Label 2-1805-1.1-c1-0-55
Degree $2$
Conductor $1805$
Sign $-1$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.23·2-s + 3.00·4-s − 5-s − 2·7-s − 2.23·8-s − 3·9-s + 2.23·10-s + 4.47·13-s + 4.47·14-s − 0.999·16-s − 2·17-s + 6.70·18-s − 3.00·20-s + 6·23-s + 25-s − 10.0·26-s − 6.00·28-s + 8.94·29-s − 8.94·31-s + 6.70·32-s + 4.47·34-s + 2·35-s − 9.00·36-s + 4.47·37-s + 2.23·40-s + 8.94·41-s − 6·43-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.50·4-s − 0.447·5-s − 0.755·7-s − 0.790·8-s − 9-s + 0.707·10-s + 1.24·13-s + 1.19·14-s − 0.249·16-s − 0.485·17-s + 1.58·18-s − 0.670·20-s + 1.25·23-s + 0.200·25-s − 1.96·26-s − 1.13·28-s + 1.66·29-s − 1.60·31-s + 1.18·32-s + 0.766·34-s + 0.338·35-s − 1.50·36-s + 0.735·37-s + 0.353·40-s + 1.39·41-s − 0.914·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
19 \( 1 \)
good2 \( 1 + 2.23T + 2T^{2} \)
3 \( 1 + 3T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 8.94T + 29T^{2} \)
31 \( 1 + 8.94T + 31T^{2} \)
37 \( 1 - 4.47T + 37T^{2} \)
41 \( 1 - 8.94T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + 4.47T + 53T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 - 8.94T + 89T^{2} \)
97 \( 1 + 13.4T + 97T^{2} \)
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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

−9.014339022682654525788748670100, −8.268810609789432089132763144540, −7.59114460523762221657093265967, −6.60416628536008257449352409647, −6.12713122160770754987246398695, −4.79290576208334763260373758918, −3.48293467799567444191004825750, −2.65466888321512335124647307280, −1.20098488214232486959183582723, 0, 1.20098488214232486959183582723, 2.65466888321512335124647307280, 3.48293467799567444191004825750, 4.79290576208334763260373758918, 6.12713122160770754987246398695, 6.60416628536008257449352409647, 7.59114460523762221657093265967, 8.268810609789432089132763144540, 9.014339022682654525788748670100

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