Properties

Label 2-1805-1.1-c1-0-21
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 $0$

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: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.504775435\)
\(L(\frac12)\) \(\approx\) \(1.504775435\)
\(L(\frac{3}{2})\) 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

−9.180269078376159410057067247792, −8.641046451413056059928979018312, −7.888671076690515591144389934644, −7.06260344577378957035420993766, −6.11223643147922350650706661543, −5.09286214690838306892529783179, −3.83978926023296414656361190013, −3.53808712262848033914246425072, −2.62075476335234995502737434357, −0.795184987854384961401443682104, 0.795184987854384961401443682104, 2.62075476335234995502737434357, 3.53808712262848033914246425072, 3.83978926023296414656361190013, 5.09286214690838306892529783179, 6.11223643147922350650706661543, 7.06260344577378957035420993766, 7.888671076690515591144389934644, 8.641046451413056059928979018312, 9.180269078376159410057067247792

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