Properties

Label 1-920-920.733-r0-0-0
Degree $1$
Conductor $920$
Sign $-0.0313 + 0.999i$
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  + (−0.755 + 0.654i)3-s + (0.909 + 0.415i)7-s + (0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.909 − 0.415i)13-s + (−0.540 + 0.841i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + (0.540 + 0.841i)27-s + (0.841 + 0.540i)29-s + (−0.654 + 0.755i)31-s + (0.909 − 0.415i)33-s + (0.989 + 0.142i)37-s + (−0.415 + 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)3-s + (0.909 + 0.415i)7-s + (0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.909 − 0.415i)13-s + (−0.540 + 0.841i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + (0.540 + 0.841i)27-s + (0.841 + 0.540i)29-s + (−0.654 + 0.755i)31-s + (0.909 − 0.415i)33-s + (0.989 + 0.142i)37-s + (−0.415 + 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0313 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0313 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.0313 + 0.999i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ -0.0313 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7263069040 + 0.7494260914i\)
\(L(\frac12)\) \(\approx\) \(0.7263069040 + 0.7494260914i\)
\(L(1)\) \(\approx\) \(0.8318551517 + 0.2786165971i\)
\(L(1)\) \(\approx\) \(0.8318551517 + 0.2786165971i\)

Euler product

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

−21.60674750958920460837049228171, −20.983801241409481003336392941, −20.1328636532521912191516766423, −19.171346055879033059578066541192, −18.17481233645321746302873044504, −17.984867623430294054475296130159, −17.03900737603827115375534228649, −16.23096500573511373584114143928, −15.438720349063033484263366760989, −14.35183398983689358802999913503, −13.432028803385889086266426639872, −13.00902557405355643522241666781, −11.864360635836954060816543747100, −11.08736936479983371283277955324, −10.73854213334970523255866846902, −9.47806477319065353684232879379, −8.25802229793100165660529568769, −7.68584394405253962824350313407, −6.749078845357353978899688854493, −5.94062691335456995777186305349, −4.84990548689736457221618284973, −4.33683617175668745243710726659, −2.63072911824483320500776133611, −1.76169190291850415924730313460, −0.57992382205922620357513001302, 1.13248173882581627791225831256, 2.40017602061655532920374251265, 3.69228606246914783296841265082, 4.50530858071730198371406060529, 5.520987967154757352131439415723, 5.95214984981134260127517264055, 7.169770802113805513000739588058, 8.47137835593368841804950099284, 8.74458895418532667759853597308, 10.339046813858076793361276071857, 10.62254904723327017907508336487, 11.416070122009969104011773167363, 12.36199877317444928543502367690, 13.1000099123172005684164357388, 14.25819069604466947179288234498, 15.15483792914105861174948956649, 15.655281538740817017599454554388, 16.51157432341738026247455421976, 17.37611818346633869138712778651, 18.10640548853432168661934866472, 18.58878130094980126198697393066, 19.88309077179542182417435517177, 20.8190897303367315585093354589, 21.33495066984050153118359413610, 21.87916233789430477971700428873

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