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Properties

Label 1-1840-1840.1363-r0-0-0
Degree $1$
Conductor $1840$
Sign $-0.999 + 0.00698i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.999 + 0.00698i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ -0.999 + 0.00698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002743881711 + 0.7857851735i\)
\(L(\frac12)\) \(\approx\) \(0.002743881711 + 0.7857851735i\)
\(L(1)\) \(\approx\) \(0.9715093429 + 0.3857497858i\)
\(L(1)\) \(\approx\) \(0.9715093429 + 0.3857497858i\)

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

−20.00616121613648521261036695926, −18.88243262829215300572495133254, −18.58023773226769885943167771923, −17.615873553412721806782828816200, −16.97333143074688721859970709185, −15.93818895736164298502047200596, −15.31514622906648365802686478067, −14.55484859144096460797478034370, −13.6831849301087210084579089879, −13.1353751409516202511686874842, −12.74562726734944367357664307776, −11.52511486229105136260701230409, −10.72747801090475421305288522509, −9.95032746597841978744467701533, −9.12606211993779023243811686288, −8.339674829411678373274499313754, −7.62477833135676934057055469392, −6.82941990596886832339522212822, −6.30829581332977597234958650077, −4.94248783786225427659211926923, −3.98328447026874167968387974614, −3.43681806099615061672971625852, −2.21689410669601923816841113768, −1.63276673814805546275667442728, −0.21575764433635691331923938463, 1.71040845411030811649083851654, 2.50751929895528046617414920004, 3.28930184885347268435075823462, 4.05699329189662925279463189845, 5.12409916487494831740781220202, 5.8232879437909847570424452399, 6.76681035051538066109437196054, 7.88934525188382717878083836902, 8.6891711204029021539017314912, 8.90073031077740616982272522924, 9.940238556742179210089992722039, 10.85505907736636151029090896080, 11.24061165577407227565576870453, 12.67528973298205479777490785661, 13.094470371470827097121468038495, 13.81659621473087020018875625320, 14.84350962525846228778214592518, 15.301336001973634750726342870206, 16.0156119511289792517710370253, 16.493596307731675494977321627294, 17.76532351820655363773649645695, 18.46756266761363110275315426020, 19.08247610552484803071347149615, 19.754404538708238850343712951431, 20.6252443908843495611053294377

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