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Properties

Label 2-960-960.629-c0-0-0
Degree $2$
Conductor $960$
Sign $0.881 - 0.471i$
Analytic cond. $0.479102$
Root an. cond. $0.692172$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.831 − 0.555i)2-s + (−0.980 + 0.195i)3-s + (0.382 + 0.923i)4-s + (0.555 + 0.831i)5-s + (0.923 + 0.382i)6-s + (0.195 − 0.980i)8-s + (0.923 − 0.382i)9-s − i·10-s + (−0.555 − 0.831i)12-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)16-s + (0.785 − 0.785i)17-s + (−0.980 − 0.195i)18-s + (−0.324 − 0.216i)19-s + (−0.555 + 0.831i)20-s + ⋯
L(s)  = 1  + (−0.831 − 0.555i)2-s + (−0.980 + 0.195i)3-s + (0.382 + 0.923i)4-s + (0.555 + 0.831i)5-s + (0.923 + 0.382i)6-s + (0.195 − 0.980i)8-s + (0.923 − 0.382i)9-s − i·10-s + (−0.555 − 0.831i)12-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)16-s + (0.785 − 0.785i)17-s + (−0.980 − 0.195i)18-s + (−0.324 − 0.216i)19-s + (−0.555 + 0.831i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :0),\ 0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5633085570\)
\(L(\frac12)\) \(\approx\) \(0.5633085570\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.831 + 0.555i)T \)
3 \( 1 + (0.980 - 0.195i)T \)
5 \( 1 + (-0.555 - 0.831i)T \)
good7 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.923 + 0.382i)T^{2} \)
13 \( 1 + (0.382 + 0.923i)T^{2} \)
17 \( 1 + (-0.785 + 0.785i)T - iT^{2} \)
19 \( 1 + (0.324 + 0.216i)T + (0.382 + 0.923i)T^{2} \)
23 \( 1 + (-0.636 - 1.53i)T + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.923 - 0.382i)T^{2} \)
31 \( 1 - 0.765iT - T^{2} \)
37 \( 1 + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.923 - 0.382i)T^{2} \)
47 \( 1 + (-1.38 + 1.38i)T - iT^{2} \)
53 \( 1 + (0.360 - 1.81i)T + (-0.923 - 0.382i)T^{2} \)
59 \( 1 + (0.382 - 0.923i)T^{2} \)
61 \( 1 + (0.382 - 0.0761i)T + (0.923 - 0.382i)T^{2} \)
67 \( 1 + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (1 + i)T + iT^{2} \)
83 \( 1 + (-1.53 - 1.02i)T + (0.382 + 0.923i)T^{2} \)
89 \( 1 + (0.707 + 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.44962847862682044303771310985, −9.601705524758622060245368740273, −9.044713987991844106858042146294, −7.52766514209565026159078558523, −7.11482739472531021321192228959, −6.11089198829179025433546590814, −5.18799122950868329483236083859, −3.81167825204261864475170281853, −2.78847861580325377275382927568, −1.36507989036505711871277021616, 0.935864577919110723324970379796, 2.13349022541472722125959876556, 4.34460785101279142741306946355, 5.25734134235439586665093601918, 5.98759864803307841254874746924, 6.62914582419739990277336667158, 7.70872541846262633005002070318, 8.466286843253908714713983970327, 9.336223730049960122066925914546, 10.20112166021110365662864065613

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