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Properties

Label 2-330-55.8-c1-0-3
Degree $2$
Conductor $330$
Sign $0.956 - 0.292i$
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 + (2.22 + 0.169i)5-s + (0.587 − 0.809i)6-s + (2.21 − 1.12i)7-s + (−0.891 − 0.453i)8-s + (−0.587 − 0.809i)9-s + (−2.17 − 0.515i)10-s + (−0.283 + 3.30i)11-s + (−0.707 + 0.707i)12-s + (0.668 − 4.22i)13-s + (−2.36 + 0.768i)14-s + (−1.16 + 1.90i)15-s + (0.809 + 0.587i)16-s + (−0.562 − 3.55i)17-s + ⋯
L(s)  = 1  + (−0.698 − 0.110i)2-s + (−0.262 + 0.514i)3-s + (0.475 + 0.154i)4-s + (0.997 + 0.0755i)5-s + (0.239 − 0.330i)6-s + (0.837 − 0.426i)7-s + (−0.315 − 0.160i)8-s + (−0.195 − 0.269i)9-s + (−0.688 − 0.163i)10-s + (−0.0853 + 0.996i)11-s + (−0.204 + 0.204i)12-s + (0.185 − 1.17i)13-s + (−0.632 + 0.205i)14-s + (−0.300 + 0.493i)15-s + (0.202 + 0.146i)16-s + (−0.136 − 0.861i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.956 - 0.292i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ 0.956 - 0.292i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12525 + 0.168507i\)
\(L(\frac12)\) \(\approx\) \(1.12525 + 0.168507i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 - 0.169i)T \)
11 \( 1 + (0.283 - 3.30i)T \)
good7 \( 1 + (-2.21 + 1.12i)T + (4.11 - 5.66i)T^{2} \)
13 \( 1 + (-0.668 + 4.22i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (0.562 + 3.55i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (-0.625 - 1.92i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-4.88 - 4.88i)T + 23iT^{2} \)
29 \( 1 + (2.39 - 7.37i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.54 + 2.57i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.20 - 2.36i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (-1.25 + 0.407i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-6.14 + 6.14i)T - 43iT^{2} \)
47 \( 1 + (3.44 + 1.75i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (13.0 + 2.06i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (6.61 + 2.14i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.07 + 1.47i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (9.79 - 9.79i)T - 67iT^{2} \)
71 \( 1 + (-8.56 - 6.22i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (6.17 + 12.1i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (3.18 - 2.31i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.92 + 0.780i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + 9.99iT - 89T^{2} \)
97 \( 1 + (0.208 - 1.31i)T + (-92.2 - 29.9i)T^{2} \)
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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

−11.23167021876489791627887877870, −10.60880888574234207138272630125, −9.805605404453982169499040031505, −9.114281924046610227796153890212, −7.85908551247320688830552435341, −6.98924764695250771535172162000, −5.62094411370598796510696609066, −4.78697764682462029754290882247, −3.03204268657045080562919968464, −1.45120117357624371024335190540, 1.37261004598896175214484904727, 2.55204167678280633424681529475, 4.74662996324326774434527044493, 5.99534568550123623989577847578, 6.54614535787836724701271667546, 7.927511696481548474750930306176, 8.743443801068251901413014107312, 9.470021953559265942418797676870, 10.83687382892178993293354165824, 11.22712689446586208120647498395

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