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Properties

Label 1-1840-1840.1539-r0-0-0
Degree $1$
Conductor $1840$
Sign $-0.685 + 0.727i$
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  + (−0.281 − 0.959i)3-s + (−0.142 − 0.989i)7-s + (−0.841 + 0.540i)9-s + (−0.909 − 0.415i)11-s + (0.989 + 0.142i)13-s + (−0.654 − 0.755i)17-s + (−0.755 − 0.654i)19-s + (−0.909 + 0.415i)21-s + (0.755 + 0.654i)27-s + (0.755 − 0.654i)29-s + (0.959 + 0.281i)31-s + (−0.142 + 0.989i)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 + (−0.142 − 0.989i)7-s + (−0.841 + 0.540i)9-s + (−0.909 − 0.415i)11-s + (0.989 + 0.142i)13-s + (−0.654 − 0.755i)17-s + (−0.755 − 0.654i)19-s + (−0.909 + 0.415i)21-s + (0.755 + 0.654i)27-s + (0.755 − 0.654i)29-s + (0.959 + 0.281i)31-s + (−0.142 + 0.989i)33-s + (−0.540 − 0.841i)37-s + (−0.142 − 0.989i)39-s + (−0.841 − 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.685 + 0.727i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ -0.685 + 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2058782049 - 0.4767844430i\)
\(L(\frac12)\) \(\approx\) \(-0.2058782049 - 0.4767844430i\)
\(L(1)\) \(\approx\) \(0.6383595576 - 0.4179681937i\)
\(L(1)\) \(\approx\) \(0.6383595576 - 0.4179681937i\)

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 \( 1 + (-0.142 - 0.989i)T \)
11 \( 1 + (-0.909 - 0.415i)T \)
13 \( 1 + (0.989 + 0.142i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (-0.755 - 0.654i)T \)
29 \( 1 + (0.755 - 0.654i)T \)
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 \( 1 + T \)
53 \( 1 + (-0.989 + 0.142i)T \)
59 \( 1 + (-0.989 - 0.142i)T \)
61 \( 1 + (-0.281 + 0.959i)T \)
67 \( 1 + (0.909 - 0.415i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (-0.654 + 0.755i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (-0.540 - 0.841i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−20.75374997318271220369491143503, −20.067586960751558410852415355354, −19.0473699797590811184181650558, −18.360664485328469591159084901369, −17.65541278285013032888050982854, −16.890021041843302819191647870459, −15.986478131466988732459794093424, −15.45965464047846620809863531936, −15.06951611333706786914643992637, −14.07332929430560030734505695976, −13.0740680318696820736963902119, −12.39529610123675932726884098064, −11.58779075067795803137885796514, −10.72158368601873741575353952692, −10.248719733442302937698081053861, −9.35462282595126684107060750277, −8.48329600484182379128393324976, −8.12156853815316144089379280049, −6.45343001671785549534439001536, −6.10633613103132742682266368654, −5.10113118708425852916334672384, −4.48953444304259057065366328364, −3.43899270308847691806567158676, −2.70198862509252569421044436979, −1.60392680664300694279047657395, 0.20460411138111427154838267921, 1.082352485909441657606470003566, 2.233275534169360476701949790831, 3.041495875149134061795988988622, 4.16315889990847994249132572702, 5.06090951507845108113896243694, 6.02454290659917308383795159679, 6.76190782960185100767821715602, 7.33509631306488514628262379406, 8.29185216205291633104381614975, 8.80951527177762421207696467922, 10.137349558640995090329585614144, 10.857793114444811765905996524, 11.337876874951390873348916571449, 12.34901066624399372494791935912, 13.13050477181880097425006247699, 13.77109732195609633259074054393, 13.987323382841955869542925035822, 15.53122011782364583445703708457, 15.93980385929137789725041604640, 17.00687500710803678804310803796, 17.44145980751427600917103113530, 18.27771768677346151605796236192, 18.88120557028592352492073136395, 19.58784527772179620901974927074

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