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Properties

Label 2-960-192.131-c1-0-19
Degree $2$
Conductor $960$
Sign $-0.0487 - 0.998i$
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  + (−1.40 − 0.164i)2-s + (0.232 − 1.71i)3-s + (1.94 + 0.462i)4-s + (−0.555 + 0.831i)5-s + (−0.609 + 2.37i)6-s + (1.58 + 3.81i)7-s + (−2.65 − 0.970i)8-s + (−2.89 − 0.797i)9-s + (0.917 − 1.07i)10-s + (−0.451 + 0.0898i)11-s + (1.24 − 3.23i)12-s + (2.44 − 1.63i)13-s + (−1.59 − 5.62i)14-s + (1.29 + 1.14i)15-s + (3.57 + 1.80i)16-s + (−2.92 + 2.92i)17-s + ⋯
L(s)  = 1  + (−0.993 − 0.116i)2-s + (0.134 − 0.990i)3-s + (0.972 + 0.231i)4-s + (−0.248 + 0.371i)5-s + (−0.248 + 0.968i)6-s + (0.597 + 1.44i)7-s + (−0.939 − 0.343i)8-s + (−0.963 − 0.265i)9-s + (0.290 − 0.340i)10-s + (−0.136 + 0.0270i)11-s + (0.359 − 0.933i)12-s + (0.679 − 0.453i)13-s + (−0.425 − 1.50i)14-s + (0.335 + 0.296i)15-s + (0.893 + 0.450i)16-s + (−0.709 + 0.709i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0487 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0487 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.0487 - 0.998i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.0487 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.389744 + 0.409227i\)
\(L(\frac12)\) \(\approx\) \(0.389744 + 0.409227i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.164i)T \)
3 \( 1 + (-0.232 + 1.71i)T \)
5 \( 1 + (0.555 - 0.831i)T \)
good7 \( 1 + (-1.58 - 3.81i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.451 - 0.0898i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (-2.44 + 1.63i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (2.92 - 2.92i)T - 17iT^{2} \)
19 \( 1 + (4.86 - 3.24i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (-1.67 + 4.03i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.38 - 6.94i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + 8.19T + 31T^{2} \)
37 \( 1 + (2.73 - 4.09i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (8.17 + 3.38i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (0.254 - 0.0506i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (-1.06 - 1.06i)T + 47iT^{2} \)
53 \( 1 + (-1.04 - 5.26i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (-4.37 - 2.92i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (-0.165 + 0.830i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-13.7 - 2.73i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (0.629 - 0.260i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (9.16 + 3.79i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (2.92 + 2.92i)T + 79iT^{2} \)
83 \( 1 + (-7.11 - 10.6i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (5.17 - 2.14i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 - 1.75iT - 97T^{2} \)
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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

−10.41521904012907757882686109638, −8.809392718666422651501607326745, −8.716805461335594250222221398625, −7.988477318651461965210132677695, −6.95893375255012065357158611115, −6.23096068318982494900065343646, −5.43352861690204925197766614262, −3.50296070163059192602563806673, −2.38350280469758833808836058081, −1.64747765866336341974308064611, 0.33973714680443731351141004492, 1.97977972356694670201968652762, 3.55701037414103443559697257853, 4.39258541889212133862552796059, 5.38189213104226564972578189880, 6.67521884267917707856410732205, 7.45941469537003462451227203302, 8.359333970392966889351326913891, 8.975686282723546711834549426569, 9.765699193682067810527614275233

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