Properties

Label 2-960-4.3-c2-0-8
Degree $2$
Conductor $960$
Sign $-i$
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  − 1.73i·3-s + 2.23·5-s + 12.3i·7-s − 2.99·9-s − 11.0i·11-s − 2.82·13-s − 3.87i·15-s + 6.52·17-s + 27.9i·19-s + 21.4·21-s − 7.90i·23-s + 5.00·25-s + 5.19i·27-s − 50.7·29-s + 36.3i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.447·5-s + 1.77i·7-s − 0.333·9-s − 1.00i·11-s − 0.216·13-s − 0.258i·15-s + 0.383·17-s + 1.47i·19-s + 1.02·21-s − 0.343i·23-s + 0.200·25-s + 0.192i·27-s − 1.74·29-s + 1.17i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ -i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.410888993\)
\(L(\frac12)\) \(\approx\) \(1.410888993\)
\(L(2)\) 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 \( 1 + 1.73iT \)
5 \( 1 - 2.23T \)
good7 \( 1 - 12.3iT - 49T^{2} \)
11 \( 1 + 11.0iT - 121T^{2} \)
13 \( 1 + 2.82T + 169T^{2} \)
17 \( 1 - 6.52T + 289T^{2} \)
19 \( 1 - 27.9iT - 361T^{2} \)
23 \( 1 + 7.90iT - 529T^{2} \)
29 \( 1 + 50.7T + 841T^{2} \)
31 \( 1 - 36.3iT - 961T^{2} \)
37 \( 1 - 18.9T + 1.36e3T^{2} \)
41 \( 1 - 5.30T + 1.68e3T^{2} \)
43 \( 1 - 45.5iT - 1.84e3T^{2} \)
47 \( 1 - 11.7iT - 2.20e3T^{2} \)
53 \( 1 + 41.1T + 2.80e3T^{2} \)
59 \( 1 + 10.7iT - 3.48e3T^{2} \)
61 \( 1 + 56.1T + 3.72e3T^{2} \)
67 \( 1 - 16.1iT - 4.48e3T^{2} \)
71 \( 1 - 66.1iT - 5.04e3T^{2} \)
73 \( 1 - 15.6T + 5.32e3T^{2} \)
79 \( 1 - 123. iT - 6.24e3T^{2} \)
83 \( 1 - 99.6iT - 6.88e3T^{2} \)
89 \( 1 - 101.T + 7.92e3T^{2} \)
97 \( 1 - 127.T + 9.40e3T^{2} \)
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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

−9.871428742368771243935546437705, −9.099855011934606121292471270710, −8.411151064451254140888649774318, −7.68403009892286831703919132244, −6.32204685380424805149878347878, −5.84878610994513432272670456202, −5.15574402449726317194699002316, −3.45354241644570808814705670596, −2.50470021392958276982795259266, −1.48015874928725388434319849530, 0.43934344230141729695294494565, 1.95737981992521532780769062645, 3.40322607350227051921871235410, 4.33016264915314912948820604274, 4.99838271768430770360414666142, 6.21981602749088586538347914895, 7.31661470617735489495553984951, 7.61292339444256579440032980840, 9.150314378139553545475890723533, 9.645935301146921008581985896439

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