Properties

Label 2-960-3.2-c2-0-49
Degree $2$
Conductor $960$
Sign $0.599 + 0.800i$
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  + (−2.40 + 1.79i)3-s + 2.23i·5-s + 10.2·7-s + (2.53 − 8.63i)9-s − 8.19i·11-s + 13.5·13-s + (−4.02 − 5.36i)15-s − 15.4i·17-s − 25.4·19-s + (−24.5 + 18.3i)21-s − 17.9i·23-s − 5.00·25-s + (9.44 + 25.2i)27-s − 42.0i·29-s − 38.4·31-s + ⋯
L(s)  = 1  + (−0.800 + 0.599i)3-s + 0.447i·5-s + 1.45·7-s + (0.281 − 0.959i)9-s − 0.744i·11-s + 1.04·13-s + (−0.268 − 0.357i)15-s − 0.910i·17-s − 1.34·19-s + (−1.16 + 0.874i)21-s − 0.778i·23-s − 0.200·25-s + (0.349 + 0.936i)27-s − 1.44i·29-s − 1.24·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.599 + 0.800i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.599 + 0.800i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.357859599\)
\(L(\frac12)\) \(\approx\) \(1.357859599\)
\(L(2)\) 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 \( 1 + (2.40 - 1.79i)T \)
5 \( 1 - 2.23iT \)
good7 \( 1 - 10.2T + 49T^{2} \)
11 \( 1 + 8.19iT - 121T^{2} \)
13 \( 1 - 13.5T + 169T^{2} \)
17 \( 1 + 15.4iT - 289T^{2} \)
19 \( 1 + 25.4T + 361T^{2} \)
23 \( 1 + 17.9iT - 529T^{2} \)
29 \( 1 + 42.0iT - 841T^{2} \)
31 \( 1 + 38.4T + 961T^{2} \)
37 \( 1 + 11.8T + 1.36e3T^{2} \)
41 \( 1 + 46.3iT - 1.68e3T^{2} \)
43 \( 1 + 54.0T + 1.84e3T^{2} \)
47 \( 1 - 43.0iT - 2.20e3T^{2} \)
53 \( 1 + 82.7iT - 2.80e3T^{2} \)
59 \( 1 + 45.8iT - 3.48e3T^{2} \)
61 \( 1 - 93.6T + 3.72e3T^{2} \)
67 \( 1 - 34.4T + 4.48e3T^{2} \)
71 \( 1 - 68.0iT - 5.04e3T^{2} \)
73 \( 1 + 44.7T + 5.32e3T^{2} \)
79 \( 1 - 11.7T + 6.24e3T^{2} \)
83 \( 1 + 144. iT - 6.88e3T^{2} \)
89 \( 1 - 63.7iT - 7.92e3T^{2} \)
97 \( 1 - 63.9T + 9.40e3T^{2} \)
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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

−9.919557931321692109013425232173, −8.750625553707540034890658762790, −8.236156969566155836525918821919, −7.01416857741814806386326017301, −6.15071074670008154416194215235, −5.34099626632700522055706318500, −4.43991987818660862541268509908, −3.57825257643865041598062550748, −2.01154852746216825360464602513, −0.50535040380761754001450683914, 1.37857402159267403906595163841, 1.88022969817211649777293691101, 3.92992262559809397192969474095, 4.85106684005577961296801195910, 5.54584834107296520823428172746, 6.52590811785008789238060008195, 7.44509681642269079803234819020, 8.274090700887095480698816723206, 8.826461125828569432085782778921, 10.25381581537781296057125428065

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