Properties

Label 2-320-80.27-c1-0-0
Degree $2$
Conductor $320$
Sign $-0.105 - 0.994i$
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  − 1.96·3-s + (−1.42 − 1.72i)5-s + (1.60 + 1.60i)7-s + 0.851·9-s + (−0.754 + 0.754i)11-s + 5.94i·13-s + (2.79 + 3.38i)15-s + (1.95 + 1.95i)17-s + (−0.780 + 0.780i)19-s + (−3.14 − 3.14i)21-s + (−4.93 + 4.93i)23-s + (−0.956 + 4.90i)25-s + 4.21·27-s + (−1.44 − 1.44i)29-s + 3.60i·31-s + ⋯
L(s)  = 1  − 1.13·3-s + (−0.635 − 0.771i)5-s + (0.605 + 0.605i)7-s + 0.283·9-s + (−0.227 + 0.227i)11-s + 1.64i·13-s + (0.720 + 0.874i)15-s + (0.474 + 0.474i)17-s + (−0.179 + 0.179i)19-s + (−0.686 − 0.686i)21-s + (−1.02 + 1.02i)23-s + (−0.191 + 0.981i)25-s + 0.811·27-s + (−0.268 − 0.268i)29-s + 0.648i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.105 - 0.994i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.105 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.370303 + 0.411597i\)
\(L(\frac12)\) \(\approx\) \(0.370303 + 0.411597i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 + 1.72i)T \)
good3 \( 1 + 1.96T + 3T^{2} \)
7 \( 1 + (-1.60 - 1.60i)T + 7iT^{2} \)
11 \( 1 + (0.754 - 0.754i)T - 11iT^{2} \)
13 \( 1 - 5.94iT - 13T^{2} \)
17 \( 1 + (-1.95 - 1.95i)T + 17iT^{2} \)
19 \( 1 + (0.780 - 0.780i)T - 19iT^{2} \)
23 \( 1 + (4.93 - 4.93i)T - 23iT^{2} \)
29 \( 1 + (1.44 + 1.44i)T + 29iT^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 - 10.2iT - 37T^{2} \)
41 \( 1 + 6.93iT - 41T^{2} \)
43 \( 1 + 9.91iT - 43T^{2} \)
47 \( 1 + (0.104 - 0.104i)T - 47iT^{2} \)
53 \( 1 + 4.03T + 53T^{2} \)
59 \( 1 + (-3.46 - 3.46i)T + 59iT^{2} \)
61 \( 1 + (-0.680 + 0.680i)T - 61iT^{2} \)
67 \( 1 - 9.04iT - 67T^{2} \)
71 \( 1 - 3.64T + 71T^{2} \)
73 \( 1 + (2.94 + 2.94i)T + 73iT^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 4.23T + 83T^{2} \)
89 \( 1 - 0.0426T + 89T^{2} \)
97 \( 1 + (1.91 + 1.91i)T + 97iT^{2} \)
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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

−11.89920671154250037037499299298, −11.32950457009820379925438871065, −10.15610276823732529240642064897, −8.970937094551269898120429467572, −8.174710870156946724906348462872, −6.97657273947851703312028989917, −5.77575997328514084510818646789, −5.00443646359248676940071172940, −3.97722880222003028260017831831, −1.69408532289789952078395533352, 0.46434136903488373139828597067, 2.93971134100686182953873611210, 4.35035839528706209602024411318, 5.49289750678743328764150694271, 6.39442261507817503371376756824, 7.60689032558301287893876369866, 8.164018404537016423693842628215, 9.945251890226173407676726575126, 10.83421511985559487318727931423, 11.10911444596315035794152007711

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