Properties

Label 2-960-20.19-c2-0-44
Degree $2$
Conductor $960$
Sign $-0.654 + 0.755i$
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 + (3.27 − 3.77i)5-s − 9.55·7-s + 2.99·9-s − 9.92i·11-s + 7.55i·13-s + (5.67 − 6.54i)15-s + 17.1i·17-s − 26.1i·19-s − 16.5·21-s + 1.67·23-s + (−3.54 − 24.7i)25-s + 5.19·27-s + 0.350·29-s − 46.0i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.654 − 0.755i)5-s − 1.36·7-s + 0.333·9-s − 0.902i·11-s + 0.581i·13-s + (0.378 − 0.436i)15-s + 1.01i·17-s − 1.37i·19-s − 0.788·21-s + 0.0728·23-s + (−0.141 − 0.989i)25-s + 0.192·27-s + 0.0120·29-s − 1.48i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.654 + 0.755i$
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: \((2,\ 960,\ (\ :1),\ -0.654 + 0.755i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.415680291\)
\(L(\frac12)\) \(\approx\) \(1.415680291\)
\(L(2)\) 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 \( 1 + (-3.27 + 3.77i)T \)
good7 \( 1 + 9.55T + 49T^{2} \)
11 \( 1 + 9.92iT - 121T^{2} \)
13 \( 1 - 7.55iT - 169T^{2} \)
17 \( 1 - 17.1iT - 289T^{2} \)
19 \( 1 + 26.1iT - 361T^{2} \)
23 \( 1 - 1.67T + 529T^{2} \)
29 \( 1 - 0.350T + 841T^{2} \)
31 \( 1 + 46.0iT - 961T^{2} \)
37 \( 1 + 22.6iT - 1.36e3T^{2} \)
41 \( 1 + 77.2T + 1.68e3T^{2} \)
43 \( 1 + 41.7T + 1.84e3T^{2} \)
47 \( 1 + 14.0T + 2.20e3T^{2} \)
53 \( 1 + 22.6iT - 2.80e3T^{2} \)
59 \( 1 + 94.7iT - 3.48e3T^{2} \)
61 \( 1 + 38T + 3.72e3T^{2} \)
67 \( 1 + 29.8T + 4.48e3T^{2} \)
71 \( 1 - 7.19iT - 5.04e3T^{2} \)
73 \( 1 - 34.3iT - 5.32e3T^{2} \)
79 \( 1 - 46.0iT - 6.24e3T^{2} \)
83 \( 1 - 24.1T + 6.88e3T^{2} \)
89 \( 1 - 100.T + 7.92e3T^{2} \)
97 \( 1 - 131. iT - 9.40e3T^{2} \)
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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

−9.448620514679004092419652091457, −8.833943184329076697101537714829, −8.108294310645175620239935838485, −6.75435692656009307073068749408, −6.23878174848819450629284981161, −5.18066519581018952283851599399, −4.00085748818378785444668807247, −3.07373298256087246620365470848, −1.91940676045167751169790719887, −0.38814877374767682492841676367, 1.69725631328386866928097046140, 2.96198889266008129213025067785, 3.42380788583507495960345614882, 4.91140165183307853716732443304, 6.01338291814520598123951121236, 6.82707877948849577259185419185, 7.41073315166320810766993754284, 8.580454647714336838157534800465, 9.534938660079355877592139527049, 10.07040393847998904103459892242

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