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Properties

Label 2-960-192.35-c1-0-11
Degree $2$
Conductor $960$
Sign $-0.0126 - 0.999i$
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  + (−1.14 − 0.836i)2-s + (−1.69 − 0.372i)3-s + (0.601 + 1.90i)4-s + (0.831 + 0.555i)5-s + (1.61 + 1.83i)6-s + (1.37 + 3.32i)7-s + (0.908 − 2.67i)8-s + (2.72 + 1.26i)9-s + (−0.483 − 1.32i)10-s + (0.927 + 4.66i)11-s + (−0.307 − 3.45i)12-s + (−0.139 − 0.208i)13-s + (1.21 − 4.94i)14-s + (−1.19 − 1.24i)15-s + (−3.27 + 2.29i)16-s + (−0.486 + 0.486i)17-s + ⋯
L(s)  = 1  + (−0.806 − 0.591i)2-s + (−0.976 − 0.215i)3-s + (0.300 + 0.953i)4-s + (0.371 + 0.248i)5-s + (0.660 + 0.750i)6-s + (0.520 + 1.25i)7-s + (0.321 − 0.946i)8-s + (0.907 + 0.420i)9-s + (−0.152 − 0.420i)10-s + (0.279 + 1.40i)11-s + (−0.0886 − 0.996i)12-s + (−0.0387 − 0.0579i)13-s + (0.323 − 1.32i)14-s + (−0.309 − 0.322i)15-s + (−0.819 + 0.573i)16-s + (−0.117 + 0.117i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0126 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0126 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.0126 - 0.999i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.0126 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.485144 + 0.491301i\)
\(L(\frac12)\) \(\approx\) \(0.485144 + 0.491301i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14 + 0.836i)T \)
3 \( 1 + (1.69 + 0.372i)T \)
5 \( 1 + (-0.831 - 0.555i)T \)
good7 \( 1 + (-1.37 - 3.32i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.927 - 4.66i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (0.139 + 0.208i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (0.486 - 0.486i)T - 17iT^{2} \)
19 \( 1 + (-2.52 - 3.77i)T + (-7.27 + 17.5i)T^{2} \)
23 \( 1 + (-0.746 + 1.80i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (5.44 + 1.08i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + 5.21T + 31T^{2} \)
37 \( 1 + (7.41 + 4.95i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (0.503 + 0.208i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.25 + 6.31i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 + (-7.39 - 7.39i)T + 47iT^{2} \)
53 \( 1 + (6.90 - 1.37i)T + (48.9 - 20.2i)T^{2} \)
59 \( 1 + (6.02 - 9.01i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (-12.7 - 2.52i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (0.654 - 3.29i)T + (-61.8 - 25.6i)T^{2} \)
71 \( 1 + (-9.82 + 4.07i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (3.16 + 1.31i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-6.49 - 6.49i)T + 79iT^{2} \)
83 \( 1 + (-0.476 + 0.318i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (6.88 - 2.85i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + 18.0iT - 97T^{2} \)
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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

−10.26070922872664019942756828776, −9.528257501161646217629496631550, −8.806045362888269731448918324561, −7.62473297313044152330432340869, −7.06063237804232163361783001823, −5.94680118590447901005452151065, −5.12872370866794117289963681985, −3.92495690246421949759911710241, −2.27655717754561235530269191138, −1.62835254091729568148150189587, 0.50541308361012115446597594306, 1.49907954142526142881338908057, 3.64716868457021117781763741894, 4.93333341957289128547219583042, 5.50944968536025522726204176610, 6.55070063222236963350538063480, 7.16070698478259160480201740304, 8.083290837617950453708656456102, 9.106286201741633275503062786632, 9.759070131171792065563228395543

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