Properties

Label 2-960-12.11-c1-0-11
Degree $2$
Conductor $960$
Sign $0.985 - 0.169i$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
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  + (−1.70 + 0.292i)3-s + i·5-s − 0.585i·7-s + (2.82 − i)9-s − 2.82·11-s + 2·13-s + (−0.292 − 1.70i)15-s − 3.65i·17-s − 2.82i·19-s + (0.171 + i)21-s + 4.58·23-s − 25-s + (−4.53 + 2.53i)27-s + 8i·29-s + 5.65i·31-s + ⋯
L(s)  = 1  + (−0.985 + 0.169i)3-s + 0.447i·5-s − 0.221i·7-s + (0.942 − 0.333i)9-s − 0.852·11-s + 0.554·13-s + (−0.0756 − 0.440i)15-s − 0.886i·17-s − 0.648i·19-s + (0.0374 + 0.218i)21-s + 0.956·23-s − 0.200·25-s + (−0.872 + 0.487i)27-s + 1.48i·29-s + 1.01i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.985 - 0.169i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.985 - 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08729 + 0.0925987i\)
\(L(\frac12)\) \(\approx\) \(1.08729 + 0.0925987i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.70 - 0.292i)T \)
5 \( 1 - iT \)
good7 \( 1 + 0.585iT - 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 3.65iT - 17T^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 - 4.58T + 23T^{2} \)
29 \( 1 - 8iT - 29T^{2} \)
31 \( 1 - 5.65iT - 31T^{2} \)
37 \( 1 - 11.6T + 37T^{2} \)
41 \( 1 - 2iT - 41T^{2} \)
43 \( 1 + 11.8iT - 43T^{2} \)
47 \( 1 - 2.24T + 47T^{2} \)
53 \( 1 + 7.65iT - 53T^{2} \)
59 \( 1 - 9.65T + 59T^{2} \)
61 \( 1 - 5.65T + 61T^{2} \)
67 \( 1 - 13.0iT - 67T^{2} \)
71 \( 1 - 6.82T + 71T^{2} \)
73 \( 1 - 4.34T + 73T^{2} \)
79 \( 1 - 12.4iT - 79T^{2} \)
83 \( 1 - 9.07T + 83T^{2} \)
89 \( 1 - 15.3iT - 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−10.24994718925563255928730577089, −9.394232645211362928548210615620, −8.394026820497932386790050152984, −7.07988302790223246056144120900, −6.89046630475133300534130116360, −5.54592286500107858042304277805, −5.01807871026151360302052969832, −3.83298226078509975248661279788, −2.64195822731942701521079276501, −0.865214052208346090838523450650, 0.889344124165243472400658211134, 2.31458868077247472819404367954, 3.94635010344571889670603757365, 4.81733059445413063157327450310, 5.85668004846700883932640289113, 6.23699734120006272994674980479, 7.62464895451334996590191240642, 8.110404819899600024099433735861, 9.310247012749419130747726189770, 10.10055248540965057508858184739

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