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Properties

Label 2-330-11.5-c1-0-2
Degree $2$
Conductor $330$
Sign $0.569 - 0.821i$
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 + (−2.92 + 2.12i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + 0.999·10-s + (2.80 + 1.76i)11-s − 0.999·12-s + (0.190 + 0.587i)13-s + (2.92 + 2.12i)14-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (−2 + 6.15i)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 + (−1.10 + 0.803i)7-s + (0.286 + 0.207i)8-s + (0.103 + 0.317i)9-s + 0.316·10-s + (0.846 + 0.531i)11-s − 0.288·12-s + (0.0529 + 0.163i)13-s + (0.782 + 0.568i)14-s + (−0.208 + 0.151i)15-s + (0.0772 − 0.237i)16-s + (−0.485 + 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.569 - 0.821i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ 0.569 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.958574 + 0.501766i\)
\(L(\frac12)\) \(\approx\) \(0.958574 + 0.501766i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.80 - 1.76i)T \)
good7 \( 1 + (2.92 - 2.12i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.190 - 0.587i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2 - 6.15i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.5 - 0.363i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 4.85T + 23T^{2} \)
29 \( 1 + (3 - 2.17i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.85 + 8.78i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.5 + 0.363i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.54 - 5.48i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.763T + 43T^{2} \)
47 \( 1 + (-2.11 - 1.53i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.88 + 5.79i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-11.8 + 8.59i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.47 + 10.6i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 8.18T + 67T^{2} \)
71 \( 1 + (0.0901 - 0.277i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.85 - 6.43i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.38 + 4.25i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.52 - 4.70i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 5.14T + 89T^{2} \)
97 \( 1 + (-4.61 - 14.2i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.59413061571420049825052095424, −10.79710088336248777278825693620, −9.655669059989283797589445777499, −9.272405866489757467741848825313, −8.251002328712578811775991471655, −6.94143971051410552197560675887, −5.89804784780482129226976420180, −4.22178400337649435252292403797, −3.32298179684728011458523645838, −2.10345506665790916986133972724, 0.813687963307333050675143980761, 3.13086424221597933270291302556, 4.30370389690286502705347162788, 5.73343853687572639578449749684, 6.96188555587856360435547733711, 7.31594958748375272133553820104, 8.846000733199955569161908569963, 9.172188544631958255440938746409, 10.27625810143771124384472904550, 11.45938714280140272697625829431

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