Properties

Label 2-320-5.4-c1-0-9
Degree $2$
Conductor $320$
Sign $-1$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.23i·3-s − 2.23·5-s − 0.763i·7-s − 7.47·9-s + 7.23i·15-s − 2.47·21-s − 5.70i·23-s + 5.00·25-s + 14.4i·27-s − 6·29-s + 1.70i·35-s − 4.47·41-s − 11.2i·43-s + 16.7·45-s − 13.7i·47-s + ⋯
L(s)  = 1  − 1.86i·3-s − 0.999·5-s − 0.288i·7-s − 2.49·9-s + 1.86i·15-s − 0.539·21-s − 1.19i·23-s + 1.00·25-s + 2.78i·27-s − 1.11·29-s + 0.288i·35-s − 0.698·41-s − 1.71i·43-s + 2.49·45-s − 1.99i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-1$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.765918i\)
\(L(\frac12)\) \(\approx\) \(0.765918i\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.23iT - 3T^{2} \)
7 \( 1 + 0.763iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 5.70iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 + 11.2iT - 43T^{2} \)
47 \( 1 + 13.7iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 + 8.18iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 17.7iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.52973018239200456695984912183, −10.55075035709947792383124374864, −8.799248625798489187961820827836, −8.145333263117380176796508711167, −7.22137510876829682312366512447, −6.70028884169748980067068633430, −5.37405656422761824414783060441, −3.68197503671921374509558661383, −2.21309292232618562980695288057, −0.54203674593918975402133761281, 3.07259212614449354672858705353, 3.96393813181188993381959928447, 4.86724775767359282115239252340, 5.88291129271952006489432174005, 7.55268921339908238736794427242, 8.595704134806719994888197460077, 9.375059098659536273399526608005, 10.20543804354717127961279603811, 11.28380129977894010022732637833, 11.54081180536531209042938298204

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