Properties

Label 2-320-4.3-c2-0-1
Degree $2$
Conductor $320$
Sign $-1$
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  + 3.80i·3-s − 2.23·5-s + 8.50i·7-s − 5.47·9-s + 1.79i·11-s − 0.472·13-s − 8.50i·15-s − 23.8·17-s − 9.40i·19-s − 32.3·21-s + 16.1i·23-s + 5.00·25-s + 13.4i·27-s − 6.94·29-s − 47.4i·31-s + ⋯
L(s)  = 1  + 1.26i·3-s − 0.447·5-s + 1.21i·7-s − 0.608·9-s + 0.163i·11-s − 0.0363·13-s − 0.567i·15-s − 1.40·17-s − 0.494i·19-s − 1.54·21-s + 0.700i·23-s + 0.200·25-s + 0.497i·27-s − 0.239·29-s − 1.53i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-1$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.969692i\)
\(L(\frac12)\) \(\approx\) \(0.969692i\)
\(L(2)\) 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 \( 1 + 2.23T \)
good3 \( 1 - 3.80iT - 9T^{2} \)
7 \( 1 - 8.50iT - 49T^{2} \)
11 \( 1 - 1.79iT - 121T^{2} \)
13 \( 1 + 0.472T + 169T^{2} \)
17 \( 1 + 23.8T + 289T^{2} \)
19 \( 1 + 9.40iT - 361T^{2} \)
23 \( 1 - 16.1iT - 529T^{2} \)
29 \( 1 + 6.94T + 841T^{2} \)
31 \( 1 + 47.4iT - 961T^{2} \)
37 \( 1 + 26.3T + 1.36e3T^{2} \)
41 \( 1 + 41.4T + 1.68e3T^{2} \)
43 \( 1 + 2.00iT - 1.84e3T^{2} \)
47 \( 1 - 35.3iT - 2.20e3T^{2} \)
53 \( 1 - 21.6T + 2.80e3T^{2} \)
59 \( 1 - 73.8iT - 3.48e3T^{2} \)
61 \( 1 - 26.1T + 3.72e3T^{2} \)
67 \( 1 - 88.8iT - 4.48e3T^{2} \)
71 \( 1 - 39.4iT - 5.04e3T^{2} \)
73 \( 1 - 137.T + 5.32e3T^{2} \)
79 \( 1 - 113. iT - 6.24e3T^{2} \)
83 \( 1 - 21.2iT - 6.88e3T^{2} \)
89 \( 1 - 67.4T + 7.92e3T^{2} \)
97 \( 1 + 39.1T + 9.40e3T^{2} \)
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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

−11.59151071566430929221383156025, −11.04900608914055974496259867117, −9.895683943238428710400436239266, −9.147279943720264490584316953772, −8.451578601633214421155859003268, −7.05420268709369504086731114474, −5.72102826704691937217778603235, −4.76683187146510693734183114139, −3.80240239140930554536661662757, −2.41804115093982427079582370547, 0.44881164330531768164790096116, 1.87010954949121341654638922819, 3.57646773042693015747548752794, 4.80447769971993259211372658725, 6.51963253962385560731232422003, 6.98799499083442646050864203757, 7.917032670545448728508925336146, 8.762734025003559397603287348353, 10.25262410122465889351417669922, 11.00561450322235681643308199020

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