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Properties

Label 2-960-16.5-c3-0-34
Degree $2$
Conductor $960$
Sign $0.538 - 0.842i$
Analytic cond. $56.6418$
Root an. cond. $7.52607$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.12 + 2.12i)3-s + (3.53 + 3.53i)5-s + 26.0i·7-s − 8.99i·9-s + (36.7 + 36.7i)11-s + (18.6 − 18.6i)13-s − 15·15-s + 130.·17-s + (89.4 − 89.4i)19-s + (−55.2 − 55.2i)21-s − 143. i·23-s + 25.0i·25-s + (19.0 + 19.0i)27-s + (91.0 − 91.0i)29-s + 26.1·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.316 + 0.316i)5-s + 1.40i·7-s − 0.333i·9-s + (1.00 + 1.00i)11-s + (0.398 − 0.398i)13-s − 0.258·15-s + 1.86·17-s + (1.08 − 1.08i)19-s + (−0.574 − 0.574i)21-s − 1.30i·23-s + 0.200i·25-s + (0.136 + 0.136i)27-s + (0.582 − 0.582i)29-s + 0.151·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.538 - 0.842i$
Analytic conductor: \(56.6418\)
Root analytic conductor: \(7.52607\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ 0.538 - 0.842i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.473531365\)
\(L(\frac12)\) \(\approx\) \(2.473531365\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.12 - 2.12i)T \)
5 \( 1 + (-3.53 - 3.53i)T \)
good7 \( 1 - 26.0iT - 343T^{2} \)
11 \( 1 + (-36.7 - 36.7i)T + 1.33e3iT^{2} \)
13 \( 1 + (-18.6 + 18.6i)T - 2.19e3iT^{2} \)
17 \( 1 - 130.T + 4.91e3T^{2} \)
19 \( 1 + (-89.4 + 89.4i)T - 6.85e3iT^{2} \)
23 \( 1 + 143. iT - 1.21e4T^{2} \)
29 \( 1 + (-91.0 + 91.0i)T - 2.43e4iT^{2} \)
31 \( 1 - 26.1T + 2.97e4T^{2} \)
37 \( 1 + (-47.9 - 47.9i)T + 5.06e4iT^{2} \)
41 \( 1 - 32.1iT - 6.89e4T^{2} \)
43 \( 1 + (96.9 + 96.9i)T + 7.95e4iT^{2} \)
47 \( 1 - 434.T + 1.03e5T^{2} \)
53 \( 1 + (-19.3 - 19.3i)T + 1.48e5iT^{2} \)
59 \( 1 + (-156. - 156. i)T + 2.05e5iT^{2} \)
61 \( 1 + (16.1 - 16.1i)T - 2.26e5iT^{2} \)
67 \( 1 + (-666. + 666. i)T - 3.00e5iT^{2} \)
71 \( 1 + 985. iT - 3.57e5T^{2} \)
73 \( 1 + 227. iT - 3.89e5T^{2} \)
79 \( 1 + 188.T + 4.93e5T^{2} \)
83 \( 1 + (899. - 899. i)T - 5.71e5iT^{2} \)
89 \( 1 - 378. iT - 7.04e5T^{2} \)
97 \( 1 - 741.T + 9.12e5T^{2} \)
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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

−9.704348978629216843579032552454, −9.182372584180711726992205441425, −8.219225820734685385940249715264, −7.09996929925341260520491413825, −6.19781207833182873778240569035, −5.49334646371886027857824335663, −4.65294548290362981250095769760, −3.35317311568239354151754549028, −2.39360166997343686745917935462, −0.979545687961951494467189894739, 1.09617294335307723270708081589, 1.19911754891803478819669257559, 3.36184330799072651542273603965, 3.97421870739011025449125746444, 5.39317558618097120694011121322, 5.97939059037869558909213915817, 7.04622230839117920850753218381, 7.68289637770748439669036093828, 8.594454085931719883114155142583, 9.729287727069946058339026926211

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