Properties

Label 2-960-3.2-c2-0-41
Degree $2$
Conductor $960$
Sign $0.970 - 0.240i$
Analytic cond. $26.1581$
Root an. cond. $5.11449$
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  + (0.721 + 2.91i)3-s + 2.23i·5-s + 1.03·7-s + (−7.95 + 4.20i)9-s − 19.5i·11-s + 16.3·13-s + (−6.51 + 1.61i)15-s − 27.1i·17-s + 24.3·19-s + (0.747 + 3.01i)21-s + 19.9i·23-s − 5.00·25-s + (−17.9 − 20.1i)27-s − 50.0i·29-s + 32.9·31-s + ⋯
L(s)  = 1  + (0.240 + 0.970i)3-s + 0.447i·5-s + 0.148·7-s + (−0.884 + 0.466i)9-s − 1.77i·11-s + 1.25·13-s + (−0.434 + 0.107i)15-s − 1.59i·17-s + 1.28·19-s + (0.0356 + 0.143i)21-s + 0.866i·23-s − 0.200·25-s + (−0.665 − 0.746i)27-s − 1.72i·29-s + 1.06·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.970 - 0.240i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.970 - 0.240i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.160650952\)
\(L(\frac12)\) \(\approx\) \(2.160650952\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.721 - 2.91i)T \)
5 \( 1 - 2.23iT \)
good7 \( 1 - 1.03T + 49T^{2} \)
11 \( 1 + 19.5iT - 121T^{2} \)
13 \( 1 - 16.3T + 169T^{2} \)
17 \( 1 + 27.1iT - 289T^{2} \)
19 \( 1 - 24.3T + 361T^{2} \)
23 \( 1 - 19.9iT - 529T^{2} \)
29 \( 1 + 50.0iT - 841T^{2} \)
31 \( 1 - 32.9T + 961T^{2} \)
37 \( 1 - 0.942T + 1.36e3T^{2} \)
41 \( 1 + 11.5iT - 1.68e3T^{2} \)
43 \( 1 - 53.0T + 1.84e3T^{2} \)
47 \( 1 - 68.3iT - 2.20e3T^{2} \)
53 \( 1 - 48.4iT - 2.80e3T^{2} \)
59 \( 1 + 14.3iT - 3.48e3T^{2} \)
61 \( 1 + 34.4T + 3.72e3T^{2} \)
67 \( 1 - 9.54T + 4.48e3T^{2} \)
71 \( 1 + 10.9iT - 5.04e3T^{2} \)
73 \( 1 - 14.7T + 5.32e3T^{2} \)
79 \( 1 + 81.0T + 6.24e3T^{2} \)
83 \( 1 + 58.1iT - 6.88e3T^{2} \)
89 \( 1 - 37.4iT - 7.92e3T^{2} \)
97 \( 1 + 96.7T + 9.40e3T^{2} \)
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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

−9.725634456668123131256273688027, −9.188300863667354485404155177931, −8.265678833775960041266769568879, −7.57030780143599984126922012026, −6.11505608783874874247540093193, −5.63695378790813221034528519764, −4.45258468860424158850698385204, −3.36143257376020253680761690336, −2.84506199388205164288765140542, −0.802246662169803549170736953755, 1.17052964293948886700981120876, 1.94671736494811158129128569247, 3.35526508900465196450100687599, 4.48127754311578316188405161317, 5.57583288867224060103740851406, 6.53151758824243328255238615113, 7.25164913959088138099728817397, 8.199669457003110954083913222144, 8.717984252354796666067263982640, 9.720332191854287339735314374159

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