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Properties

Label 1-920-920.269-r0-0-0
Degree $1$
Conductor $920$
Sign $0.270 - 0.962i$
Analytic cond. $4.27246$
Root an. cond. $4.27246$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.415 − 0.909i)3-s + (−0.841 + 0.540i)7-s + (−0.654 − 0.755i)9-s + (0.142 − 0.989i)11-s + (0.841 + 0.540i)13-s + (0.959 + 0.281i)17-s + (0.959 − 0.281i)19-s + (0.142 + 0.989i)21-s + (−0.959 + 0.281i)27-s + (0.959 + 0.281i)29-s + (0.415 + 0.909i)31-s + (−0.841 − 0.540i)33-s + (−0.654 − 0.755i)37-s + (0.841 − 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)3-s + (−0.841 + 0.540i)7-s + (−0.654 − 0.755i)9-s + (0.142 − 0.989i)11-s + (0.841 + 0.540i)13-s + (0.959 + 0.281i)17-s + (0.959 − 0.281i)19-s + (0.142 + 0.989i)21-s + (−0.959 + 0.281i)27-s + (0.959 + 0.281i)29-s + (0.415 + 0.909i)31-s + (−0.841 − 0.540i)33-s + (−0.654 − 0.755i)37-s + (0.841 − 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.270 - 0.962i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.270 - 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.278396913 - 0.9687938713i\)
\(L(\frac12)\) \(\approx\) \(1.278396913 - 0.9687938713i\)
\(L(1)\) \(\approx\) \(1.124156384 - 0.3994265799i\)
\(L(1)\) \(\approx\) \(1.124156384 - 0.3994265799i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (-0.841 + 0.540i)T \)
11 \( 1 + (0.142 - 0.989i)T \)
13 \( 1 + (0.841 + 0.540i)T \)
17 \( 1 + (0.959 + 0.281i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (0.959 + 0.281i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (-0.654 - 0.755i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 - T \)
53 \( 1 + (0.841 - 0.540i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (-0.415 - 0.909i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (0.959 - 0.281i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.654 - 0.755i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.15201695858996717402995935426, −21.00750559173350482920802790689, −20.544205439371901911082984192319, −19.874069641751581123558223716, −19.072778738784837585327009087253, −18.074265554890614619382642961122, −17.102658183421541212223609164559, −16.36965394352529207689675639244, −15.69255035897980348956061892307, −15.02417941410513521000367458999, −13.99480878070554930670591030520, −13.46198421132811779281941342804, −12.38409077363940069926630909386, −11.49349933934799124184854466317, −10.25066179336516812810642772453, −10.03750059879920085113667956431, −9.16962606796645781370100516166, −8.114397129238063823266569122592, −7.33106914817165771762954221368, −6.19361200529694187778278266823, −5.243011731991985627643548536709, −4.243138469085608500687128583820, −3.44721702473101307750664619952, −2.702372057171407020392646264028, −1.15101360878088977160396534547, 0.79560719598118384637443627970, 1.84687623023610797926399361667, 3.2325376385549723007325099826, 3.39693409810474526518883244572, 5.20824440644594156596993896005, 6.17310370255237285353880551995, 6.65836867743270569019870856055, 7.77891427165203454650095149487, 8.65590290493494202113874009923, 9.18708833679438196409315817767, 10.29736794291876518698911545302, 11.46942540081200937245724177384, 12.12501483414408130845995270027, 12.88564852654235814477795591781, 13.8476093187488877048527607561, 14.14089116166583447375841091838, 15.423204869735763869091766625588, 16.1482023632052989440880415744, 16.922738679303872361238608203052, 18.126828392996867205436052070701, 18.55958755040547691058106285524, 19.38278188599410630584373229953, 19.802806132242700367609888795289, 21.018372124761407915457206089610, 21.5833107770356548127755126024

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