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Properties

Label 2-330-33.2-c1-0-6
Degree $2$
Conductor $330$
Sign $0.870 - 0.492i$
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 + (1.63 − 0.557i)3-s + (−0.809 − 0.587i)4-s + (−0.951 + 0.309i)5-s + (0.0236 + 1.73i)6-s + (0.325 − 0.447i)7-s + (0.809 − 0.587i)8-s + (2.37 − 1.82i)9-s − 0.999i·10-s + (3.02 + 1.34i)11-s + (−1.65 − 0.512i)12-s + (3.86 + 1.25i)13-s + (0.325 + 0.447i)14-s + (−1.38 + 1.03i)15-s + (0.309 + 0.951i)16-s + (0.413 + 1.27i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.946 − 0.321i)3-s + (−0.404 − 0.293i)4-s + (−0.425 + 0.138i)5-s + (0.00964 + 0.707i)6-s + (0.122 − 0.169i)7-s + (0.286 − 0.207i)8-s + (0.792 − 0.609i)9-s − 0.316i·10-s + (0.913 + 0.406i)11-s + (−0.477 − 0.148i)12-s + (1.07 + 0.348i)13-s + (0.0869 + 0.119i)14-s + (−0.358 + 0.267i)15-s + (0.0772 + 0.237i)16-s + (0.100 + 0.308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.870 - 0.492i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ 0.870 - 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52018 + 0.400358i\)
\(L(\frac12)\) \(\approx\) \(1.52018 + 0.400358i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 + 0.557i)T \)
5 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (-3.02 - 1.34i)T \)
good7 \( 1 + (-0.325 + 0.447i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-3.86 - 1.25i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.413 - 1.27i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.719 - 0.990i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.15iT - 23T^{2} \)
29 \( 1 + (4.24 + 3.08i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.446 - 1.37i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (7.78 + 5.65i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.54 - 2.57i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.30iT - 43T^{2} \)
47 \( 1 + (3.79 + 5.22i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (5.63 + 1.83i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.90 - 8.12i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-10.4 + 3.38i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 15.2T + 67T^{2} \)
71 \( 1 + (-1.34 + 0.437i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.17 + 4.36i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.97 - 1.61i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.12 + 9.61i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 6.44iT - 89T^{2} \)
97 \( 1 + (0.00726 - 0.0223i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.73420273812636153507636169677, −10.56556834705450449776303591642, −9.449982325811711632253469981589, −8.714948020137482400616566142307, −7.894208642220868213659437782433, −6.98891894518701715289724028859, −6.16211755925023330533958954214, −4.40590578211772146864341666112, −3.50938978424323039278157782878, −1.56726453174187243534219127866, 1.55300632935233046609128691708, 3.24241942656978518057992747750, 3.90289544722475382458671803193, 5.27774295696481748685685608112, 6.94736647152440292235463166318, 8.117149415622058967656301775274, 8.795254045088348326807694916269, 9.503662055399639683934841522738, 10.61248677349808943997785757143, 11.40551632899818665827352040081

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