Properties

Label 2-3330-1.1-c1-0-8
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 − 4·7-s + 8-s − 10-s − 6·11-s + 2·13-s − 4·14-s + 16-s + 6·17-s + 2·19-s − 20-s − 6·22-s + 25-s + 2·26-s − 4·28-s − 6·29-s + 8·31-s + 32-s + 6·34-s + 4·35-s + 37-s + 2·38-s − 40-s + 6·41-s + 8·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s − 1.80·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s − 0.223·20-s − 1.27·22-s + 1/5·25-s + 0.392·26-s − 0.755·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.676·35-s + 0.164·37-s + 0.324·38-s − 0.158·40-s + 0.937·41-s + 1.21·43-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

\(L(1)\) \(\approx\) \(1.937164869\)
\(L(\frac12)\) \(\approx\) \(1.937164869\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−8.424912579227892857567266604331, −7.71588865532144785570929233268, −7.18162370286094383162742724296, −6.14700327202829166226347590269, −5.67324598464159891720762446042, −4.85568846993463928899190098959, −3.73420480399354799573660891311, −3.17995762137668904356640145254, −2.47135784523752124203055675221, −0.72114804858626419919238278961, 0.72114804858626419919238278961, 2.47135784523752124203055675221, 3.17995762137668904356640145254, 3.73420480399354799573660891311, 4.85568846993463928899190098959, 5.67324598464159891720762446042, 6.14700327202829166226347590269, 7.18162370286094383162742724296, 7.71588865532144785570929233268, 8.424912579227892857567266604331

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