Properties

Label 2-3330-1.1-c1-0-3
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 − 3·7-s − 8-s + 10-s + 5·11-s − 2·13-s + 3·14-s + 16-s − 3·17-s − 6·19-s − 20-s − 5·22-s + 4·23-s + 25-s + 2·26-s − 3·28-s + 29-s − 3·31-s − 32-s + 3·34-s + 3·35-s − 37-s + 6·38-s + 40-s + 7·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.13·7-s − 0.353·8-s + 0.316·10-s + 1.50·11-s − 0.554·13-s + 0.801·14-s + 1/4·16-s − 0.727·17-s − 1.37·19-s − 0.223·20-s − 1.06·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.566·28-s + 0.185·29-s − 0.538·31-s − 0.176·32-s + 0.514·34-s + 0.507·35-s − 0.164·37-s + 0.973·38-s + 0.158·40-s + 1.09·41-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

\(L(1)\) \(\approx\) \(0.8152152977\)
\(L(\frac12)\) \(\approx\) \(0.8152152977\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−8.929198720137352528743349049695, −7.936020171375730953473914631474, −6.99879135181147946789051897764, −6.62470444155780626408199418622, −5.95780477626079696380170828616, −4.61551401185348386625175209239, −3.86686308710105270428125307275, −2.98677838409226059495247173192, −1.94000625275420551426618824942, −0.58281538244965167502065518669, 0.58281538244965167502065518669, 1.94000625275420551426618824942, 2.98677838409226059495247173192, 3.86686308710105270428125307275, 4.61551401185348386625175209239, 5.95780477626079696380170828616, 6.62470444155780626408199418622, 6.99879135181147946789051897764, 7.936020171375730953473914631474, 8.929198720137352528743349049695

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