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Properties

Label 1-920-920.469-r0-0-0
Degree $1$
Conductor $920$
Sign $-0.559 - 0.828i$
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  + (−0.142 + 0.989i)3-s + (0.654 + 0.755i)7-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)11-s + (−0.654 + 0.755i)13-s + (−0.415 − 0.909i)17-s + (−0.415 + 0.909i)19-s + (−0.841 + 0.540i)21-s + (0.415 − 0.909i)27-s + (−0.415 − 0.909i)29-s + (−0.142 − 0.989i)31-s + (0.654 − 0.755i)33-s + (−0.959 − 0.281i)37-s + (−0.654 − 0.755i)39-s + (−0.959 + 0.281i)41-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)3-s + (0.654 + 0.755i)7-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)11-s + (−0.654 + 0.755i)13-s + (−0.415 − 0.909i)17-s + (−0.415 + 0.909i)19-s + (−0.841 + 0.540i)21-s + (0.415 − 0.909i)27-s + (−0.415 − 0.909i)29-s + (−0.142 − 0.989i)31-s + (0.654 − 0.755i)33-s + (−0.959 − 0.281i)37-s + (−0.654 − 0.755i)39-s + (−0.959 + 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.559 - 0.828i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ -0.559 - 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04183417404 + 0.07876008155i\)
\(L(\frac12)\) \(\approx\) \(-0.04183417404 + 0.07876008155i\)
\(L(1)\) \(\approx\) \(0.6846926309 + 0.2773654837i\)
\(L(1)\) \(\approx\) \(0.6846926309 + 0.2773654837i\)

Euler product

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

−21.367038577530990236392934011242, −20.213991380049863092889272871087, −19.97310785863860713299500178355, −18.95798470474777649708181159561, −18.0725713057120648180974500846, −17.42274924493422916617201869581, −17.050812523529729492981500385145, −15.70899017474891578752035209040, −14.86547950089348126619838499966, −14.09796248858247938167599593451, −13.12014540987124283358711554035, −12.73529352424525759123051804097, −11.74303902231617502700307770725, −10.7349020164832915984635684152, −10.311762467685791535852694928557, −8.78633559607658788682384900125, −8.08073474930057053003991732306, −7.255161118699142019938394487697, −6.70914100986086191782268075202, −5.35925672524797545097408759003, −4.81278898241855975284115467423, −3.40323837972836751705613840654, −2.28165089741508392618005336203, −1.448956663058110464494766319814, −0.0370594005539996560870326780, 1.98904271998729626115039398572, 2.83470460546201440462019311779, 3.99430207256962802710481230056, 4.92644875787709525127748381446, 5.49573113550983506994419167228, 6.49989740087202427489289166635, 7.86617804704015339799705129697, 8.53482468203121828707308497348, 9.47937880174615467621862372550, 10.10816846914761041979020559916, 11.336832086898597385265742360512, 11.51504793969846910946757428914, 12.66469555883884549008117965882, 13.821119540962458346422433456265, 14.570283696344199296030937751609, 15.26800745535827247975564407825, 16.02701238249752754214811806240, 16.726532313044987648238945665798, 17.57187303968014578767839317020, 18.483932621764846438605790475686, 19.147738423545042320880545433383, 20.31041377222746282617490075575, 21.070662988507031292519483530291, 21.40549036878683237521425214232, 22.319023774503624075567717933215

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