Properties

Label 2-1805-1.1-c1-0-43
Degree $2$
Conductor $1805$
Sign $-1$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s − 5-s − 4·7-s + 9-s + 3·11-s + 4·12-s + 2·13-s + 2·15-s + 4·16-s + 6·17-s + 2·20-s + 8·21-s + 25-s + 4·27-s + 8·28-s − 3·29-s − 7·31-s − 6·33-s + 4·35-s − 2·36-s + 8·37-s − 4·39-s − 6·41-s − 4·43-s − 6·44-s − 45-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.904·11-s + 1.15·12-s + 0.554·13-s + 0.516·15-s + 16-s + 1.45·17-s + 0.447·20-s + 1.74·21-s + 1/5·25-s + 0.769·27-s + 1.51·28-s − 0.557·29-s − 1.25·31-s − 1.04·33-s + 0.676·35-s − 1/3·36-s + 1.31·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s − 0.904·44-s − 0.149·45-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: yes
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 \( 1 + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−9.127628990679212829761037204252, −8.125110666225905114287996427728, −7.15688226851931363649415056933, −6.21260132573943813797134499627, −5.78864269288415358256356915363, −4.85687055939197751337392409509, −3.76032779270688982178507540089, −3.30897413440669327666321670557, −1.06417714624828988770031977779, 0, 1.06417714624828988770031977779, 3.30897413440669327666321670557, 3.76032779270688982178507540089, 4.85687055939197751337392409509, 5.78864269288415358256356915363, 6.21260132573943813797134499627, 7.15688226851931363649415056933, 8.125110666225905114287996427728, 9.127628990679212829761037204252

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