Properties

Label 2-3330-1.1-c1-0-33
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 11-s + 2·13-s + 14-s + 16-s + 7·17-s − 2·19-s + 20-s + 22-s + 25-s + 2·26-s + 28-s − 9·29-s + 7·31-s + 32-s + 7·34-s + 35-s − 37-s − 2·38-s + 40-s + 11·41-s − 11·43-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 + 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.69·17-s − 0.458·19-s + 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.67·29-s + 1.25·31-s + 0.176·32-s + 1.20·34-s + 0.169·35-s − 0.164·37-s − 0.324·38-s + 0.158·40-s + 1.71·41-s − 1.67·43-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: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.782772470\)
\(L(\frac12)\) \(\approx\) \(3.782772470\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−8.466617633077070121388500125275, −7.86937068984361513321196063819, −7.01876672076142607453158356850, −6.18886276950982726295041374367, −5.59971120593909584633623194371, −4.87490661096650033311220182165, −3.90115203723576073672316655816, −3.20644720863003928618543113454, −2.08181925588604955993799561820, −1.14418970765477512031133450528, 1.14418970765477512031133450528, 2.08181925588604955993799561820, 3.20644720863003928618543113454, 3.90115203723576073672316655816, 4.87490661096650033311220182165, 5.59971120593909584633623194371, 6.18886276950982726295041374367, 7.01876672076142607453158356850, 7.86937068984361513321196063819, 8.466617633077070121388500125275

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