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Properties

Label 1-1840-1840.1077-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.823 + 0.566i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  + (−0.841 + 0.540i)3-s + (−0.281 − 0.959i)7-s + (0.415 − 0.909i)9-s + (−0.755 + 0.654i)11-s + (0.959 + 0.281i)13-s + (0.989 + 0.142i)17-s + (−0.989 + 0.142i)19-s + (0.755 + 0.654i)21-s + (0.142 + 0.989i)27-s + (−0.989 − 0.142i)29-s + (0.841 + 0.540i)31-s + (0.281 − 0.959i)33-s + (0.415 − 0.909i)37-s + (−0.959 + 0.281i)39-s + (−0.415 − 0.909i)41-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)3-s + (−0.281 − 0.959i)7-s + (0.415 − 0.909i)9-s + (−0.755 + 0.654i)11-s + (0.959 + 0.281i)13-s + (0.989 + 0.142i)17-s + (−0.989 + 0.142i)19-s + (0.755 + 0.654i)21-s + (0.142 + 0.989i)27-s + (−0.989 − 0.142i)29-s + (0.841 + 0.540i)31-s + (0.281 − 0.959i)33-s + (0.415 − 0.909i)37-s + (−0.959 + 0.281i)39-s + (−0.415 − 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.823 + 0.566i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.823 + 0.566i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9393546916 + 0.2918929746i\)
\(L(\frac12)\) \(\approx\) \(0.9393546916 + 0.2918929746i\)
\(L(1)\) \(\approx\) \(0.7845081246 + 0.09504381608i\)
\(L(1)\) \(\approx\) \(0.7845081246 + 0.09504381608i\)

Euler product

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

−19.993390137273796210837601552365, −18.85321939961604482479248773215, −18.67372761363901403187301865960, −18.14809251645158890216970162939, −16.9711018382960331400183621007, −16.60441359978390139877622021257, −15.636285054551844061679989737333, −15.19177738123359509287066102022, −13.962273666735289227944939614622, −13.12948338165977687804175146842, −12.77096569536465024437269490980, −11.77570546589338152201712989728, −11.28400260810272583985575662738, −10.43042203775042580311137477150, −9.68333614940561671671579209627, −8.36856558234713332598080415060, −8.1658630383789834702765388891, −6.92397893512624094077318693990, −6.13110554940833029649013130175, −5.61829491673696948180804018650, −4.90129048753993132945987907804, −3.612520707636847850005143694418, −2.68219808964358976298747720824, −1.73825841626491143661435796401, −0.58682419114498982866700117570, 0.74119747585360016510298240651, 1.80809429950507691853475357900, 3.25262641399374728893277763405, 3.99916595148608479266901271113, 4.66964002584834740938871854812, 5.6375318957261651684042011207, 6.35109495457521274864845213336, 7.169084767522914329391254024938, 7.97762848836758920994358664500, 9.06854941371194506696643318928, 9.951849431380629696244984131119, 10.505896214467660560019544683595, 11.024130686999800273529534795057, 12.00962487624199579042358115486, 12.75930721059723862158221008951, 13.41026917950239691740688584289, 14.396235088821163268944789065797, 15.20954943615914159082759364201, 15.91770557604559578162987625625, 16.642911244595285312239507406715, 17.08596325959350834399521379377, 17.97020259426035050499642580164, 18.584074785119742802623607856989, 19.494454776055548734580270695029, 20.421347091442969435058237136

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