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Properties

Label 1-920-920.757-r0-0-0
Degree $1$
Conductor $920$
Sign $0.224 + 0.974i$
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  + (−0.281 − 0.959i)3-s + (−0.989 + 0.142i)7-s + (−0.841 + 0.540i)9-s + (0.415 − 0.909i)11-s + (−0.989 − 0.142i)13-s + (−0.755 + 0.654i)17-s + (0.654 − 0.755i)19-s + (0.415 + 0.909i)21-s + (0.755 + 0.654i)27-s + (−0.654 − 0.755i)29-s + (−0.959 − 0.281i)31-s + (−0.989 − 0.142i)33-s + (0.540 + 0.841i)37-s + (0.142 + 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯
L(s)  = 1  + (−0.281 − 0.959i)3-s + (−0.989 + 0.142i)7-s + (−0.841 + 0.540i)9-s + (0.415 − 0.909i)11-s + (−0.989 − 0.142i)13-s + (−0.755 + 0.654i)17-s + (0.654 − 0.755i)19-s + (0.415 + 0.909i)21-s + (0.755 + 0.654i)27-s + (−0.654 − 0.755i)29-s + (−0.959 − 0.281i)31-s + (−0.989 − 0.142i)33-s + (0.540 + 0.841i)37-s + (0.142 + 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.224 + 0.974i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.224 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2645262029 + 0.2106135186i\)
\(L(\frac12)\) \(\approx\) \(0.2645262029 + 0.2106135186i\)
\(L(1)\) \(\approx\) \(0.6554363037 - 0.1658979524i\)
\(L(1)\) \(\approx\) \(0.6554363037 - 0.1658979524i\)

Euler product

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

−22.0009988792369472957432640381, −20.8878115947067135327005513346, −20.05188898933722653169616454194, −19.72794408443431308261540070969, −18.460338810319821223294154991381, −17.62289290105722763017809882295, −16.769843080162988049563494535087, −16.23505603230776526521640248602, −15.41675247311250769017055897772, −14.64822729605816626540787347951, −13.89047599663490814093502602606, −12.61392881492260410497263828637, −12.12860983242776047859459681225, −11.06015801585887528468876754822, −10.23886397626497561452735583978, −9.37392212401948302264316801356, −9.16751784661725082381941262444, −7.51962124415614657077280438151, −6.82447999148819757005748180792, −5.76008946463521943283534850495, −4.89810860951806977133670763960, −4.00697084158342378157788111924, −3.18930328112988322814452308564, −2.03918154193133795785212816886, −0.16681288052468358825012326886, 1.09689020224304283630935145071, 2.43165009435596363735040918409, 3.13543231145764978497177366526, 4.45012678580383016337531082390, 5.72338428224299224353710070482, 6.25353355070135189747617316667, 7.13669337150486385350319990992, 7.93442124750962218783581809337, 9.003395188399841815168470761, 9.70160206754334918778298896165, 10.994627599522071304287612562124, 11.54604645994525530692440470086, 12.59758554064967868924627817888, 13.07444698159938861392927691243, 13.86245137764306870373046279455, 14.79087214267311959221983124815, 15.7970194411195222102760824383, 16.701519525668115468478188427908, 17.249339166859287513922889938958, 18.18427352193707775312106235268, 18.9677159304811085617993400896, 19.64683482051925523393184453507, 20.03198178458633328086033929523, 21.51956762483177682883163022825, 22.29306286726961578115840644151

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