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Properties

Label 1-920-920.27-r0-0-0
Degree $1$
Conductor $920$
Sign $-0.946 + 0.322i$
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  + (−0.540 − 0.841i)3-s + (0.281 + 0.959i)7-s + (−0.415 + 0.909i)9-s + (−0.654 − 0.755i)11-s + (0.281 − 0.959i)13-s + (−0.989 − 0.142i)17-s + (0.142 + 0.989i)19-s + (0.654 − 0.755i)21-s + (0.989 − 0.142i)27-s + (−0.142 + 0.989i)29-s + (−0.841 − 0.540i)31-s + (−0.281 + 0.959i)33-s + (−0.909 − 0.415i)37-s + (−0.959 + 0.281i)39-s + (0.415 + 0.909i)41-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)3-s + (0.281 + 0.959i)7-s + (−0.415 + 0.909i)9-s + (−0.654 − 0.755i)11-s + (0.281 − 0.959i)13-s + (−0.989 − 0.142i)17-s + (0.142 + 0.989i)19-s + (0.654 − 0.755i)21-s + (0.989 − 0.142i)27-s + (−0.142 + 0.989i)29-s + (−0.841 − 0.540i)31-s + (−0.281 + 0.959i)33-s + (−0.909 − 0.415i)37-s + (−0.959 + 0.281i)39-s + (0.415 + 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.946 + 0.322i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ -0.946 + 0.322i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01925933157 - 0.1162780571i\)
\(L(\frac12)\) \(\approx\) \(0.01925933157 - 0.1162780571i\)
\(L(1)\) \(\approx\) \(0.6614349256 - 0.1504189261i\)
\(L(1)\) \(\approx\) \(0.6614349256 - 0.1504189261i\)

Euler product

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

−22.32625941819059409254572171460, −21.48883212698520566176516160114, −20.768424179092351550800527980521, −20.19882615373073247516097129986, −19.29279129936914984072550460751, −18.03469898402652110387583739284, −17.52720153678945588320850046763, −16.811765295569777990049233473364, −15.907004160069768921947216348233, −15.36830407608425180527388611055, −14.38795459115125091285763217731, −13.56991940299268803733512723476, −12.67030543336244453827249576047, −11.52346587679170080819021050709, −11.00566473860248582509204777289, −10.23070860365553055984715947280, −9.414522793998619303604466732248, −8.57602171435316148769856285102, −7.28692632045488734605465940885, −6.67836937076313076541724323971, −5.514020435882123925290945718726, −4.460251827686913799827744239766, −4.20824375441440412692666547350, −2.847399789673222921786634113646, −1.52869199404370709725464733401, 0.05407182020441760708324404006, 1.534906121216593139321319007295, 2.44668745424141369929630533908, 3.45858811151772484171351945308, 5.08066243670714452040140096942, 5.584447630913090145075700543673, 6.35873058644364465857136975321, 7.463289044057706967043075033329, 8.26793573054999539413973278834, 8.88149522854583095660689611229, 10.31007003902028697474191060949, 11.04608840994808008703986368318, 11.76176470443363929995490768474, 12.68076749310228904968232805723, 13.183583808559712539168876712617, 14.13872535445517089729231495828, 15.117429063487707228783800787, 15.97219278966946817465062005015, 16.68859249724192677648618080090, 17.813451891331331382418187035387, 18.24739668940175293699199650465, 18.81475152813908265377944432829, 19.77380896231372675458579441237, 20.6358778643275227402206988250, 21.616659008476263693794473653903

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