Properties

Label 2-960-3.2-c2-0-45
Degree $2$
Conductor $960$
Sign $-0.745 + 0.666i$
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 − 2.23i)3-s − 2.23i·5-s + 2·7-s + (−1.00 + 8.94i)9-s + 13.4i·11-s − 8·13-s + (−5.00 + 4.47i)15-s − 13.4i·17-s + 34·19-s + (−4 − 4.47i)21-s − 40.2i·23-s − 5.00·25-s + (22.0 − 15.6i)27-s − 40.2i·29-s + 14·31-s + ⋯
L(s)  = 1  + (−0.666 − 0.745i)3-s − 0.447i·5-s + 0.285·7-s + (−0.111 + 0.993i)9-s + 1.21i·11-s − 0.615·13-s + (−0.333 + 0.298i)15-s − 0.789i·17-s + 1.78·19-s + (−0.190 − 0.212i)21-s − 1.74i·23-s − 0.200·25-s + (0.814 − 0.579i)27-s − 1.38i·29-s + 0.451·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.745 + 0.666i$
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: \((2,\ 960,\ (\ :1),\ -0.745 + 0.666i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9969387161\)
\(L(\frac12)\) \(\approx\) \(0.9969387161\)
\(L(2)\) 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 \( 1 + (2 + 2.23i)T \)
5 \( 1 + 2.23iT \)
good7 \( 1 - 2T + 49T^{2} \)
11 \( 1 - 13.4iT - 121T^{2} \)
13 \( 1 + 8T + 169T^{2} \)
17 \( 1 + 13.4iT - 289T^{2} \)
19 \( 1 - 34T + 361T^{2} \)
23 \( 1 + 40.2iT - 529T^{2} \)
29 \( 1 + 40.2iT - 841T^{2} \)
31 \( 1 - 14T + 961T^{2} \)
37 \( 1 + 56T + 1.36e3T^{2} \)
41 \( 1 - 26.8iT - 1.68e3T^{2} \)
43 \( 1 + 8T + 1.84e3T^{2} \)
47 \( 1 - 40.2iT - 2.20e3T^{2} \)
53 \( 1 + 40.2iT - 2.80e3T^{2} \)
59 \( 1 - 13.4iT - 3.48e3T^{2} \)
61 \( 1 - 46T + 3.72e3T^{2} \)
67 \( 1 + 32T + 4.48e3T^{2} \)
71 \( 1 + 53.6iT - 5.04e3T^{2} \)
73 \( 1 + 106T + 5.32e3T^{2} \)
79 \( 1 + 22T + 6.24e3T^{2} \)
83 \( 1 + 120. iT - 6.88e3T^{2} \)
89 \( 1 + 107. iT - 7.92e3T^{2} \)
97 \( 1 - 122T + 9.40e3T^{2} \)
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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

−9.705036937464569071749571404752, −8.540313096315016142133425429182, −7.60054242649656119659032606813, −7.11048432436237405698794120653, −6.09602535962335190624310227400, −4.97228266640970187732595685077, −4.61925192641444519660858165194, −2.76015719354215676684179659340, −1.66115977713216637502098421771, −0.37828556931405939858838502408, 1.26814734680581366092497485280, 3.15944751268388825804045014826, 3.73107779751504392107274286673, 5.23172816797888894396580101117, 5.52824387187321171465181983749, 6.68928295667757949413786333596, 7.54612093480690035841000551630, 8.633970359932460989212575507850, 9.456582575263715267585242820110, 10.24782610220659871149923157772

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