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Properties

Label 2-960-5.2-c2-0-27
Degree $2$
Conductor $960$
Sign $0.130 + 0.991i$
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  + (1.22 − 1.22i)3-s + (−4.67 + 1.77i)5-s + (0.550 + 0.550i)7-s − 2.99i·9-s − 1.55·11-s + (−9.55 + 9.55i)13-s + (−3.55 + 7.89i)15-s + (11.1 + 11.1i)17-s − 12.6i·19-s + 1.34·21-s + (21.3 − 21.3i)23-s + (18.6 − 16.5i)25-s + (−3.67 − 3.67i)27-s − 44.0i·29-s + 44.4·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.934 + 0.355i)5-s + (0.0786 + 0.0786i)7-s − 0.333i·9-s − 0.140·11-s + (−0.734 + 0.734i)13-s + (−0.236 + 0.526i)15-s + (0.655 + 0.655i)17-s − 0.668i·19-s + 0.0642·21-s + (0.928 − 0.928i)23-s + (0.747 − 0.663i)25-s + (−0.136 − 0.136i)27-s − 1.51i·29-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.130 + 0.991i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.130 + 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.390062596\)
\(L(\frac12)\) \(\approx\) \(1.390062596\)
\(L(2)\) 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 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (4.67 - 1.77i)T \)
good7 \( 1 + (-0.550 - 0.550i)T + 49iT^{2} \)
11 \( 1 + 1.55T + 121T^{2} \)
13 \( 1 + (9.55 - 9.55i)T - 169iT^{2} \)
17 \( 1 + (-11.1 - 11.1i)T + 289iT^{2} \)
19 \( 1 + 12.6iT - 361T^{2} \)
23 \( 1 + (-21.3 + 21.3i)T - 529iT^{2} \)
29 \( 1 + 44.0iT - 841T^{2} \)
31 \( 1 - 44.4T + 961T^{2} \)
37 \( 1 + (20.6 + 20.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 48.2T + 1.68e3T^{2} \)
43 \( 1 + (36.2 - 36.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (42.5 + 42.5i)T + 2.20e3iT^{2} \)
53 \( 1 + (-54.4 + 54.4i)T - 2.80e3iT^{2} \)
59 \( 1 + 47.4iT - 3.48e3T^{2} \)
61 \( 1 - 59.8T + 3.72e3T^{2} \)
67 \( 1 + (-81.2 - 81.2i)T + 4.48e3iT^{2} \)
71 \( 1 + 87.5T + 5.04e3T^{2} \)
73 \( 1 + (-75.9 + 75.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 97.3iT - 6.24e3T^{2} \)
83 \( 1 + (-41.0 + 41.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 52.2iT - 7.92e3T^{2} \)
97 \( 1 + (37 + 37i)T + 9.40e3iT^{2} \)
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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

−9.691741137858033650787315678220, −8.432751902102058923544688103634, −8.175672932120533677620152261338, −6.97693506249017562056894995417, −6.64032425895462716265760699729, −5.13280184757798981923963724751, −4.19918230839065001623890438774, −3.16089129563756653645597965011, −2.15138117447380221104731746403, −0.48068709561520182112917469903, 1.14307744541407147017863664111, 2.92529594646162256639181546879, 3.58609744596121770340230158755, 4.85020153068870057867500992671, 5.30304295130584260847197891458, 6.88465365547743098197119209893, 7.66466111985554167253265979292, 8.290504833226760549309737390404, 9.145870786956793083963778386204, 10.03352004163376709716692945979

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