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Properties

Label 1-920-920.893-r0-0-0
Degree $1$
Conductor $920$
Sign $0.735 + 0.677i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.735 + 0.677i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (893, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.735 + 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5526517165 + 0.2159141923i\)
\(L(\frac12)\) \(\approx\) \(0.5526517165 + 0.2159141923i\)
\(L(1)\) \(\approx\) \(0.6926695710 - 0.1426504946i\)
\(L(1)\) \(\approx\) \(0.6926695710 - 0.1426504946i\)

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

−22.011217215900032647787194546788, −20.99089511925064467422278301713, −20.43253387886631305977917790775, −19.49283931399573928806651154083, −18.50774961201998366669948305748, −17.52014995361007800105066053140, −17.352648637730718227620133020440, −15.89052858542176046203091902567, −15.418232958849694485434048322797, −15.14475805454361055953316829021, −13.75651021208232705734385355799, −12.734261693242761163803453966966, −12.1375038629239976713598304201, −11.17626964346412147916419392771, −10.42667072341425696203035632261, −9.58976028533869629548229197585, −8.926554507410869088449816083330, −7.8757882518876394184419512418, −6.70244834367327225739778375697, −5.80583253824936687670281462885, −5.0572584371046874959551549135, −4.29864022790614384597666077733, −3.00788106648781707191079674545, −2.27141955907455443647678837660, −0.31930358794801227865016994386, 1.02548011696103858023159413229, 2.08728556380324094940025852070, 3.25728591169850580337374277549, 4.419392250455175140524693505019, 5.35748380259818825202275886430, 6.42315859509271864757721118781, 6.972551789989204946284263322516, 7.86841230609898257222087113606, 8.712002405666625244125171861287, 9.94741065724523866880239808926, 10.80149712761142523836095993716, 11.415769288901419215589245975181, 12.38937712425126424148048515884, 13.18320916379007608883935452554, 13.83042297312240338798819446293, 14.488111050854435807493654563081, 16.080055703314493913669216624951, 16.35393816060756151955376126259, 17.29383651632003326676437964740, 17.99737174906605223513064033418, 18.87152262436113886591659241807, 19.44384811403989850422825447488, 20.263870186958395192828465033878, 21.25605089257419719400267872378, 22.07333430855180666540461944802

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