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Properties

Label 2-330-15.2-c1-0-15
Degree $2$
Conductor $330$
Sign $-0.852 + 0.522i$
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  + (0.707 − 0.707i)2-s + (−0.580 + 1.63i)3-s − 1.00i·4-s + (−2.01 − 0.967i)5-s + (0.743 + 1.56i)6-s + (−3.05 − 3.05i)7-s + (−0.707 − 0.707i)8-s + (−2.32 − 1.89i)9-s + (−2.10 + 0.741i)10-s + i·11-s + (1.63 + 0.580i)12-s + (0.227 − 0.227i)13-s − 4.31·14-s + (2.74 − 2.72i)15-s − 1.00·16-s + (−2.59 + 2.59i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.335 + 0.942i)3-s − 0.500i·4-s + (−0.901 − 0.432i)5-s + (0.303 + 0.638i)6-s + (−1.15 − 1.15i)7-s + (−0.250 − 0.250i)8-s + (−0.775 − 0.631i)9-s + (−0.667 + 0.234i)10-s + 0.301i·11-s + (0.471 + 0.167i)12-s + (0.0629 − 0.0629i)13-s − 1.15·14-s + (0.709 − 0.704i)15-s − 0.250·16-s + (−0.630 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.852 + 0.522i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ -0.852 + 0.522i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.154246 - 0.546791i\)
\(L(\frac12)\) \(\approx\) \(0.154246 - 0.546791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.580 - 1.63i)T \)
5 \( 1 + (2.01 + 0.967i)T \)
11 \( 1 - iT \)
good7 \( 1 + (3.05 + 3.05i)T + 7iT^{2} \)
13 \( 1 + (-0.227 + 0.227i)T - 13iT^{2} \)
17 \( 1 + (2.59 - 2.59i)T - 17iT^{2} \)
19 \( 1 + 6.02iT - 19T^{2} \)
23 \( 1 + (2.43 + 2.43i)T + 23iT^{2} \)
29 \( 1 + 5.12T + 29T^{2} \)
31 \( 1 - 9.53T + 31T^{2} \)
37 \( 1 + (5.81 + 5.81i)T + 37iT^{2} \)
41 \( 1 - 5.49iT - 41T^{2} \)
43 \( 1 + (-0.399 + 0.399i)T - 43iT^{2} \)
47 \( 1 + (3.02 - 3.02i)T - 47iT^{2} \)
53 \( 1 + (-3.88 - 3.88i)T + 53iT^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 4.44T + 61T^{2} \)
67 \( 1 + (5.84 + 5.84i)T + 67iT^{2} \)
71 \( 1 + 8.78iT - 71T^{2} \)
73 \( 1 + (2.94 - 2.94i)T - 73iT^{2} \)
79 \( 1 + 6.57iT - 79T^{2} \)
83 \( 1 + (9.42 + 9.42i)T + 83iT^{2} \)
89 \( 1 - 8.48T + 89T^{2} \)
97 \( 1 + (-0.508 - 0.508i)T + 97iT^{2} \)
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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

−11.11484077387320761443464590906, −10.45793209090558059287937809533, −9.604214933914559449338265479254, −8.656119149755124654251633335822, −7.16444155640817846082826595253, −6.18249924524927656740669481288, −4.68213803099932364435867215820, −4.10560039940463936557645532319, −3.12882136623992826124221495205, −0.34012458100311714441584114769, 2.56009030324479119644727238802, 3.66435612078413426342897014465, 5.36768574876679616158386834507, 6.26359858941918726048829181657, 6.94441626424545462785761142263, 8.015120534199694733792583596157, 8.789106222997611175522002486785, 10.19384828545511591168801650765, 11.71592725525240266194316555215, 11.85983583761192317953720313425

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