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Properties

Label 1-2340-2340.2243-r0-0-0
Degree $1$
Conductor $2340$
Sign $0.941 + 0.338i$
Analytic cond. $10.8669$
Root an. cond. $10.8669$
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  + 7-s + (0.866 + 0.5i)11-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s − i·23-s + (−0.5 + 0.866i)29-s + (0.866 + 0.5i)31-s + (0.5 + 0.866i)37-s − i·41-s − i·43-s + (0.5 + 0.866i)47-s + 49-s − i·53-s + (0.866 − 0.5i)59-s + 61-s + ⋯
L(s)  = 1  + 7-s + (0.866 + 0.5i)11-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s − i·23-s + (−0.5 + 0.866i)29-s + (0.866 + 0.5i)31-s + (0.5 + 0.866i)37-s − i·41-s − i·43-s + (0.5 + 0.866i)47-s + 49-s − i·53-s + (0.866 − 0.5i)59-s + 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.941 + 0.338i$
Analytic conductor: \(10.8669\)
Root analytic conductor: \(10.8669\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (2243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2340,\ (0:\ ),\ 0.941 + 0.338i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.064533380 + 0.3595796833i\)
\(L(\frac12)\) \(\approx\) \(2.064533380 + 0.3595796833i\)
\(L(1)\) \(\approx\) \(1.304891896 + 0.07957362296i\)
\(L(1)\) \(\approx\) \(1.304891896 + 0.07957362296i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 + T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 - iT \)
43 \( 1 - iT \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + T \)
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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

−19.480146277550618341757268921756, −18.858363714353381146116408736084, −18.178707527617842226365765415261, −17.19653226151228957407262918587, −16.96351988312349819483805879990, −16.00882251894072322021051717811, −15.08003307785168329290674018001, −14.56884474562466587327520866621, −13.859073097131082638879681372616, −13.19393242300795843811995633282, −12.01006323001440268593532063647, −11.69774908750225666901630549252, −10.91325961262617513992293993177, −10.042943866534608983320536020952, −9.26032771592646943810512975010, −8.43754010566006883623330600537, −7.796744460476496288118641001296, −7.02556546587343183890037981963, −5.94312452089743072127630392961, −5.45687222616094282976739299675, −4.31599949091066653725469053701, −3.80526831124094809399929329062, −2.646578992885255372982346575134, −1.70534780703518192763636647314, −0.85218968721563961997324783785, 1.02249324875679468103163865127, 1.79083132285974546610382567434, 2.73299680114016815359589916611, 3.87507276028497322128243328726, 4.54591696231255585000035278850, 5.27634125095828738714952968989, 6.31941864364329358304368370458, 6.95503483844736129517117809505, 7.935133070921734970421839580587, 8.52256759708117416800833676239, 9.29674734259223724666923439956, 10.25045642729963261603619255953, 10.871856109917054886141775795151, 11.69713884877245499186265254654, 12.33123907305205627477359373821, 13.0509922669285438115992817168, 14.10109030580048558629386705831, 14.66469417772303694112779651231, 15.052951361182521664204216615300, 16.14850497675070321787674745196, 16.9182321272100771161567574720, 17.46599446140913817043304518872, 18.09780995496335098360481880736, 19.01995529906058010376724471794, 19.54208300537293362865827958869

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