Properties

Label 2-960-20.19-c2-0-0
Degree $2$
Conductor $960$
Sign $-0.854 + 0.518i$
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  − 1.73·3-s + (−4.27 + 2.59i)5-s + 0.837·7-s + 2.99·9-s + 15.7i·11-s − 5.18i·13-s + (7.40 − 4.49i)15-s + 27.3i·17-s − 17.9i·19-s − 1.45·21-s − 19.1·23-s + (11.5 − 22.1i)25-s − 5.19·27-s + 45.6·29-s + 13.6i·31-s + ⋯
L(s)  = 1  − 0.577·3-s + (−0.854 + 0.518i)5-s + 0.119·7-s + 0.333·9-s + 1.43i·11-s − 0.398i·13-s + (0.493 − 0.299i)15-s + 1.60i·17-s − 0.945i·19-s − 0.0690·21-s − 0.830·23-s + (0.461 − 0.886i)25-s − 0.192·27-s + 1.57·29-s + 0.439i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.854 + 0.518i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ -0.854 + 0.518i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1239584946\)
\(L(\frac12)\) \(\approx\) \(0.1239584946\)
\(L(2)\) 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 \( 1 + 1.73T \)
5 \( 1 + (4.27 - 2.59i)T \)
good7 \( 1 - 0.837T + 49T^{2} \)
11 \( 1 - 15.7iT - 121T^{2} \)
13 \( 1 + 5.18iT - 169T^{2} \)
17 \( 1 - 27.3iT - 289T^{2} \)
19 \( 1 + 17.9iT - 361T^{2} \)
23 \( 1 + 19.1T + 529T^{2} \)
29 \( 1 - 45.6T + 841T^{2} \)
31 \( 1 - 13.6iT - 961T^{2} \)
37 \( 1 - 15.5iT - 1.36e3T^{2} \)
41 \( 1 - 13.2T + 1.68e3T^{2} \)
43 \( 1 + 27.9T + 1.84e3T^{2} \)
47 \( 1 + 55.6T + 2.20e3T^{2} \)
53 \( 1 - 15.5iT - 2.80e3T^{2} \)
59 \( 1 + 87.6iT - 3.48e3T^{2} \)
61 \( 1 + 38T + 3.72e3T^{2} \)
67 \( 1 + 92.2T + 4.48e3T^{2} \)
71 \( 1 + 130. iT - 5.04e3T^{2} \)
73 \( 1 - 54.7iT - 5.32e3T^{2} \)
79 \( 1 + 13.6iT - 6.24e3T^{2} \)
83 \( 1 + 59.0T + 6.88e3T^{2} \)
89 \( 1 - 39.8T + 7.92e3T^{2} \)
97 \( 1 + 168. iT - 9.40e3T^{2} \)
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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

−10.40643835069912489952116468135, −9.754561623288557886492261947605, −8.424228702261911872810706333997, −7.80520812661276864059058482321, −6.83724922778901603144018962154, −6.26674926908928759083998764892, −4.87684012676824255535799025813, −4.28712369632386514003786992840, −3.09347233109709529023541219935, −1.68654242179898356822947790841, 0.04857835803796854480342185229, 1.15757574447309014126204355160, 2.96625263476086775497748392040, 4.04061733237711833512104368735, 4.92571065849843056458041354143, 5.81804039668772689813040353168, 6.74066258306610225598698286739, 7.78251605864011389351525522131, 8.401132197499851691516925229971, 9.297147612017778744272863416392

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