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Properties

Label 2-330-11.2-c2-0-4
Degree $2$
Conductor $330$
Sign $0.00599 - 0.999i$
Analytic cond. $8.99184$
Root an. cond. $2.99864$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 − 0.437i)2-s + (−1.40 + 1.01i)3-s + (1.61 + 1.17i)4-s + (0.690 + 2.12i)5-s + (2.32 − 0.756i)6-s + (−0.802 + 1.10i)7-s + (−1.66 − 2.28i)8-s + (0.927 − 2.85i)9-s − 3.16i·10-s + (10.4 − 3.56i)11-s − 3.46·12-s + (8.95 + 2.90i)13-s + (1.56 − 1.13i)14-s + (−3.13 − 2.27i)15-s + (1.23 + 3.80i)16-s + (−19.1 + 6.20i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (−0.467 + 0.339i)3-s + (0.404 + 0.293i)4-s + (0.138 + 0.425i)5-s + (0.388 − 0.126i)6-s + (−0.114 + 0.157i)7-s + (−0.207 − 0.286i)8-s + (0.103 − 0.317i)9-s − 0.316i·10-s + (0.945 − 0.324i)11-s − 0.288·12-s + (0.688 + 0.223i)13-s + (0.111 − 0.0810i)14-s + (−0.208 − 0.151i)15-s + (0.0772 + 0.237i)16-s + (−1.12 + 0.365i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00599 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.00599 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.00599 - 0.999i$
Analytic conductor: \(8.99184\)
Root analytic conductor: \(2.99864\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1),\ 0.00599 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.626286 + 0.622541i\)
\(L(\frac12)\) \(\approx\) \(0.626286 + 0.622541i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.437i)T \)
3 \( 1 + (1.40 - 1.01i)T \)
5 \( 1 + (-0.690 - 2.12i)T \)
11 \( 1 + (-10.4 + 3.56i)T \)
good7 \( 1 + (0.802 - 1.10i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (-8.95 - 2.90i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (19.1 - 6.20i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-7.97 - 10.9i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + 11.6T + 529T^{2} \)
29 \( 1 + (-1.04 + 1.43i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (10.7 - 32.9i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-41.1 - 29.9i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (3.09 + 4.25i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 70.6iT - 1.84e3T^{2} \)
47 \( 1 + (70.9 - 51.5i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (8.83 - 27.1i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-29.6 - 21.5i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-61.4 + 19.9i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 120.T + 4.48e3T^{2} \)
71 \( 1 + (18.4 + 56.9i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-70.2 + 96.7i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-20.4 - 6.65i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-28.9 + 9.40i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 15.1T + 7.92e3T^{2} \)
97 \( 1 + (31.1 - 95.8i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−11.33640217089337529845632876446, −10.77089765448456420444079253798, −9.714944334889978013727838843412, −9.005049289664299862907370296272, −7.959823258683511705111654204828, −6.57751599461809635842756053643, −6.08469198300824749225723820545, −4.38576643747749452130586806461, −3.19084054132797571123445059731, −1.45971261512291417765376153605, 0.58399817853141806567055539122, 2.03916204634529903448416666447, 4.03111624767085373521346166288, 5.38902314387670861598928822228, 6.45701226491338944971405398656, 7.19580082108090062827970722088, 8.395072245397084953596809793720, 9.214040602366632940783035705951, 10.08977882154797361161526785278, 11.25871441453665179273492175904

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