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Properties

Label 2-330-55.7-c1-0-5
Degree $2$
Conductor $330$
Sign $0.670 + 0.741i$
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  + (−0.987 + 0.156i)2-s + (−0.453 − 0.891i)3-s + (0.951 − 0.309i)4-s + (−1.68 − 1.47i)5-s + (0.587 + 0.809i)6-s + (4.48 + 2.28i)7-s + (−0.891 + 0.453i)8-s + (−0.587 + 0.809i)9-s + (1.89 + 1.19i)10-s + (1.10 + 3.12i)11-s + (−0.707 − 0.707i)12-s + (−0.566 − 3.57i)13-s + (−4.78 − 1.55i)14-s + (−0.547 + 2.16i)15-s + (0.809 − 0.587i)16-s + (1.03 − 6.53i)17-s + ⋯
L(s)  = 1  + (−0.698 + 0.110i)2-s + (−0.262 − 0.514i)3-s + (0.475 − 0.154i)4-s + (−0.752 − 0.658i)5-s + (0.239 + 0.330i)6-s + (1.69 + 0.863i)7-s + (−0.315 + 0.160i)8-s + (−0.195 + 0.269i)9-s + (0.598 + 0.376i)10-s + (0.334 + 0.942i)11-s + (−0.204 − 0.204i)12-s + (−0.157 − 0.992i)13-s + (−1.27 − 0.415i)14-s + (−0.141 + 0.559i)15-s + (0.202 − 0.146i)16-s + (0.250 − 1.58i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.670 + 0.741i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ 0.670 + 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.844884 - 0.375274i\)
\(L(\frac12)\) \(\approx\) \(0.844884 - 0.375274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.987 - 0.156i)T \)
3 \( 1 + (0.453 + 0.891i)T \)
5 \( 1 + (1.68 + 1.47i)T \)
11 \( 1 + (-1.10 - 3.12i)T \)
good7 \( 1 + (-4.48 - 2.28i)T + (4.11 + 5.66i)T^{2} \)
13 \( 1 + (0.566 + 3.57i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-1.03 + 6.53i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (-1.27 + 3.93i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-4.60 + 4.60i)T - 23iT^{2} \)
29 \( 1 + (-1.62 - 5.01i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.62 + 1.90i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.809 + 1.58i)T + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (-0.559 - 0.181i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (1.68 + 1.68i)T + 43iT^{2} \)
47 \( 1 + (-2.59 + 1.32i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-1.11 + 0.176i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (-3.47 + 1.13i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.62 + 2.22i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (3.30 + 3.30i)T + 67iT^{2} \)
71 \( 1 + (8.21 - 5.96i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.05 - 2.06i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (-7.99 - 5.80i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.51 + 1.34i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 - 11.3iT - 89T^{2} \)
97 \( 1 + (-0.0938 - 0.592i)T + (-92.2 + 29.9i)T^{2} \)
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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

−11.55318849341587392092770476949, −10.76178842235635437468673864811, −9.260535263069598414010683378902, −8.593041376114213593840800543909, −7.67796460938699181762078892911, −7.08984196539176854362040716533, −5.27902129491547185973383184607, −4.83211189401928977353411312585, −2.53980532832024301650138871294, −1.02323325272924642671250545320, 1.46294258674333493402969766753, 3.55815899181569637818975245815, 4.40374196030157934406982075912, 5.93282054383196169600207886170, 7.20076370469451708778379307001, 8.007509039546305721662804325009, 8.741323651530001136173780165085, 10.13122614600868786024269809506, 10.87734638698270711208985582851, 11.38241267452723834142933267119

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