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Properties

Label 2-330-11.9-c1-0-7
Degree $2$
Conductor $330$
Sign $-0.605 + 0.795i$
Analytic cond. $2.63506$
Root an. cond. $1.62328$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (−0.309 − 0.951i)6-s + (−1 − 0.726i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s − 0.999·10-s + (−2.54 − 2.12i)11-s − 0.999·12-s + (1.73 − 5.34i)13-s + (−1 + 0.726i)14-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)16-s + (1.57 + 4.84i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.138 − 0.425i)5-s + (−0.126 − 0.388i)6-s + (−0.377 − 0.274i)7-s + (−0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s − 0.316·10-s + (−0.767 − 0.641i)11-s − 0.288·12-s + (0.481 − 1.48i)13-s + (−0.267 + 0.194i)14-s + (−0.208 − 0.151i)15-s + (0.0772 + 0.237i)16-s + (0.381 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.640549 - 1.29206i\)
\(L(\frac12)\) \(\approx\) \(0.640549 - 1.29206i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (2.54 + 2.12i)T \)
good7 \( 1 + (1 + 0.726i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.73 + 5.34i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.57 - 4.84i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.61 + 1.90i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 0.145T + 23T^{2} \)
29 \( 1 + (-0.309 - 0.224i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.11 - 3.44i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.97 - 5.06i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.61 + 1.90i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + (9.16 - 6.65i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.145 - 0.449i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.35 + 1.71i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 6.09T + 67T^{2} \)
71 \( 1 + (-0.472 - 1.45i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-10.8 - 7.88i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.19 + 6.74i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.23 - 3.80i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (3.90 - 12.0i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.16687259026203794151963031470, −10.45268419736800118357745351767, −9.496217721167916747085135789053, −8.346807098200605864266798744909, −7.79286147304935546480815734430, −6.19170422630343967325151238835, −5.20719914704941253733190087673, −3.70997624221636420694094543977, −2.83195699459143473572801186695, −0.970268837371295582242613531242, 2.51661609414376649462293063011, 3.82082453585937748784114443466, 4.91243239663821299967650822777, 6.12970518136246865667113092510, 7.21812875903029804720616546149, 7.925335912575709165486973319990, 9.274999369752186504933088313339, 9.659731456886951668417080529170, 11.00147434511321305793709598581, 11.96089375489684221339369170821

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