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Properties

Label 2-320-16.3-c2-0-13
Degree $2$
Conductor $320$
Sign $0.0986 + 0.995i$
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  + (3.48 − 3.48i)3-s + (1.58 − 1.58i)5-s + 6.27·7-s − 15.2i·9-s + (−10.5 − 10.5i)11-s + (4.99 + 4.99i)13-s − 11.0i·15-s + 16.5·17-s + (−25.9 + 25.9i)19-s + (21.8 − 21.8i)21-s + 21.6·23-s − 5.00i·25-s + (−21.7 − 21.7i)27-s + (−4.57 − 4.57i)29-s − 9.09i·31-s + ⋯
L(s)  = 1  + (1.16 − 1.16i)3-s + (0.316 − 0.316i)5-s + 0.895·7-s − 1.69i·9-s + (−0.955 − 0.955i)11-s + (0.384 + 0.384i)13-s − 0.734i·15-s + 0.973·17-s + (−1.36 + 1.36i)19-s + (1.03 − 1.03i)21-s + 0.941·23-s − 0.200i·25-s + (−0.806 − 0.806i)27-s + (−0.157 − 0.157i)29-s − 0.293i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0986 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0986 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.0986 + 0.995i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ 0.0986 + 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.94829 - 1.76469i\)
\(L(\frac12)\) \(\approx\) \(1.94829 - 1.76469i\)
\(L(2)\) 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 \( 1 + (-3.48 + 3.48i)T - 9iT^{2} \)
7 \( 1 - 6.27T + 49T^{2} \)
11 \( 1 + (10.5 + 10.5i)T + 121iT^{2} \)
13 \( 1 + (-4.99 - 4.99i)T + 169iT^{2} \)
17 \( 1 - 16.5T + 289T^{2} \)
19 \( 1 + (25.9 - 25.9i)T - 361iT^{2} \)
23 \( 1 - 21.6T + 529T^{2} \)
29 \( 1 + (4.57 + 4.57i)T + 841iT^{2} \)
31 \( 1 + 9.09iT - 961T^{2} \)
37 \( 1 + (12.8 - 12.8i)T - 1.36e3iT^{2} \)
41 \( 1 + 23.8iT - 1.68e3T^{2} \)
43 \( 1 + (36.3 + 36.3i)T + 1.84e3iT^{2} \)
47 \( 1 - 91.0iT - 2.20e3T^{2} \)
53 \( 1 + (-37.1 + 37.1i)T - 2.80e3iT^{2} \)
59 \( 1 + (-27.3 - 27.3i)T + 3.48e3iT^{2} \)
61 \( 1 + (-11.6 - 11.6i)T + 3.72e3iT^{2} \)
67 \( 1 + (-46.0 + 46.0i)T - 4.48e3iT^{2} \)
71 \( 1 - 56.7T + 5.04e3T^{2} \)
73 \( 1 - 28.1iT - 5.32e3T^{2} \)
79 \( 1 - 58.5iT - 6.24e3T^{2} \)
83 \( 1 + (38.2 - 38.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 19.9iT - 7.92e3T^{2} \)
97 \( 1 - 89.5T + 9.40e3T^{2} \)
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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

−11.28483453569539493688427695596, −10.23421757931805099636273238438, −8.880808893604205564806640622262, −8.252244577042972427249456466106, −7.72285050219395386229124987914, −6.43350504896067597041814281086, −5.32327641767165594567051772333, −3.65160174299888645772585665769, −2.33441503305625084318017219590, −1.25015564776801196383215198021, 2.15594874508735875599577878691, 3.20426199043722336950856824745, 4.54839032000986577840033888155, 5.24847270029063998874268804448, 7.05007440366387135346841081759, 8.127980061561963958927336252686, 8.787869626399279025270229597373, 9.857317053344649853393725349077, 10.48460716148525159584631497004, 11.24678415505547776347317065208

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