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Properties

Label 2-320-16.11-c2-0-15
Degree $2$
Conductor $320$
Sign $-0.809 + 0.587i$
Analytic cond. $8.71936$
Root an. cond. $2.95285$
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  + (1.39 + 1.39i)3-s + (−1.58 − 1.58i)5-s − 9.44·7-s − 5.08i·9-s + (7.42 − 7.42i)11-s + (−16.5 + 16.5i)13-s − 4.42i·15-s − 16.9·17-s + (−23.4 − 23.4i)19-s + (−13.2 − 13.2i)21-s + 0.786·23-s + 5.00i·25-s + (19.7 − 19.7i)27-s + (17.6 − 17.6i)29-s + 23.5i·31-s + ⋯
L(s)  = 1  + (0.466 + 0.466i)3-s + (−0.316 − 0.316i)5-s − 1.34·7-s − 0.565i·9-s + (0.674 − 0.674i)11-s + (−1.27 + 1.27i)13-s − 0.294i·15-s − 0.997·17-s + (−1.23 − 1.23i)19-s + (−0.629 − 0.629i)21-s + 0.0341·23-s + 0.200i·25-s + (0.729 − 0.729i)27-s + (0.607 − 0.607i)29-s + 0.758i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.809 + 0.587i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -0.809 + 0.587i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.120579 - 0.371291i\)
\(L(\frac12)\) \(\approx\) \(0.120579 - 0.371291i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.58 + 1.58i)T \)
good3 \( 1 + (-1.39 - 1.39i)T + 9iT^{2} \)
7 \( 1 + 9.44T + 49T^{2} \)
11 \( 1 + (-7.42 + 7.42i)T - 121iT^{2} \)
13 \( 1 + (16.5 - 16.5i)T - 169iT^{2} \)
17 \( 1 + 16.9T + 289T^{2} \)
19 \( 1 + (23.4 + 23.4i)T + 361iT^{2} \)
23 \( 1 - 0.786T + 529T^{2} \)
29 \( 1 + (-17.6 + 17.6i)T - 841iT^{2} \)
31 \( 1 - 23.5iT - 961T^{2} \)
37 \( 1 + (25.6 + 25.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 19.6iT - 1.68e3T^{2} \)
43 \( 1 + (-5.42 + 5.42i)T - 1.84e3iT^{2} \)
47 \( 1 - 3.62iT - 2.20e3T^{2} \)
53 \( 1 + (40.5 + 40.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (24.1 - 24.1i)T - 3.48e3iT^{2} \)
61 \( 1 + (-39.1 + 39.1i)T - 3.72e3iT^{2} \)
67 \( 1 + (24.6 + 24.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 12.9T + 5.04e3T^{2} \)
73 \( 1 - 97.1iT - 5.32e3T^{2} \)
79 \( 1 - 54.5iT - 6.24e3T^{2} \)
83 \( 1 + (-70.0 - 70.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 95.6iT - 7.92e3T^{2} \)
97 \( 1 - 97.1T + 9.40e3T^{2} \)
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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

−11.06339176276786148695931171136, −9.795156904214986709904437502926, −9.176280341905084231485698509026, −8.614255475515118008475349428736, −6.83455269963314458652033743713, −6.48702625045160099508625173829, −4.64539974079572656162074111902, −3.79880160505537213728503921288, −2.57030493304630338699121665345, −0.16082157235351376724962195665, 2.18771075047933265899216700240, 3.26961995531683476042138271370, 4.62961897945657086332294740668, 6.16872748963801611290693009654, 7.04943338530170720290746831469, 7.86391216527371734610902871638, 8.930772793204817044326444925297, 10.05161001186880820248023974257, 10.59210151870798577843457808768, 12.11585800189240137476184517555

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