Properties

Label 2-3330-1.1-c1-0-54
Degree $2$
Conductor $3330$
Sign $-1$
Analytic cond. $26.5901$
Root an. cond. $5.15656$
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-s + 4-s − 5-s + 7-s + 8-s − 10-s − 3·11-s + 14-s + 16-s − 3·17-s − 6·19-s − 20-s − 3·22-s − 2·23-s + 25-s + 28-s + 3·29-s + 3·31-s + 32-s − 3·34-s − 35-s − 37-s − 6·38-s − 40-s − 3·41-s − 43-s − 3·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.904·11-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 1.37·19-s − 0.223·20-s − 0.639·22-s − 0.417·23-s + 1/5·25-s + 0.188·28-s + 0.557·29-s + 0.538·31-s + 0.176·32-s − 0.514·34-s − 0.169·35-s − 0.164·37-s − 0.973·38-s − 0.158·40-s − 0.468·41-s − 0.152·43-s − 0.452·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $-1$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3330} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
37 \( 1 + T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−8.146451497996909742414741839962, −7.55947946358028496449425566831, −6.56405990058932844390013504676, −6.08056229568137165407273954903, −4.83332190202465636941309325541, −4.63919100492136925221940798677, −3.56154188220255317605428378165, −2.65455170884079945670816110578, −1.73957054101312727200784638700, 0, 1.73957054101312727200784638700, 2.65455170884079945670816110578, 3.56154188220255317605428378165, 4.63919100492136925221940798677, 4.83332190202465636941309325541, 6.08056229568137165407273954903, 6.56405990058932844390013504676, 7.55947946358028496449425566831, 8.146451497996909742414741839962

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