↑

Properties

Label 1-2580-2580.1187-r0-0-0
Degree $1$
Conductor $2580$
Sign $0.999 + 0.0296i$
Analytic cond. $11.9814$
Root an. cond. $11.9814$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)7-s + (0.623 + 0.781i)11-s + (−0.294 − 0.955i)13-s + (0.680 + 0.733i)17-s + (−0.365 − 0.930i)19-s + (0.149 + 0.988i)23-s + (−0.826 − 0.563i)29-s + (−0.0747 + 0.997i)31-s + (0.866 + 0.5i)37-s + (0.900 − 0.433i)41-s + (0.781 + 0.623i)47-s + (0.5 − 0.866i)49-s + (−0.294 + 0.955i)53-s + (0.222 + 0.974i)59-s + (−0.0747 − 0.997i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)7-s + (0.623 + 0.781i)11-s + (−0.294 − 0.955i)13-s + (0.680 + 0.733i)17-s + (−0.365 − 0.930i)19-s + (0.149 + 0.988i)23-s + (−0.826 − 0.563i)29-s + (−0.0747 + 0.997i)31-s + (0.866 + 0.5i)37-s + (0.900 − 0.433i)41-s + (0.781 + 0.623i)47-s + (0.5 − 0.866i)49-s + (−0.294 + 0.955i)53-s + (0.222 + 0.974i)59-s + (−0.0747 − 0.997i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2580\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 43\)
Sign: $0.999 + 0.0296i$
Analytic conductor: \(11.9814\)
Root analytic conductor: \(11.9814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2580} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2580,\ (0:\ ),\ 0.999 + 0.0296i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.006374482 + 0.02974303067i\)
\(L(\frac12)\) \(\approx\) \(2.006374482 + 0.02974303067i\)
\(L(1)\) \(\approx\) \(1.258629143 + 0.01725458701i\)
\(L(1)\) \(\approx\) \(1.258629143 + 0.01725458701i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
43 \( 1 \)
good7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (-0.294 - 0.955i)T \)
17 \( 1 + (0.680 + 0.733i)T \)
19 \( 1 + (-0.365 - 0.930i)T \)
23 \( 1 + (0.149 + 0.988i)T \)
29 \( 1 + (-0.826 - 0.563i)T \)
31 \( 1 + (-0.0747 + 0.997i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (-0.294 + 0.955i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (0.930 - 0.365i)T \)
71 \( 1 + (0.988 + 0.149i)T \)
73 \( 1 + (0.294 + 0.955i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.563 - 0.826i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
97 \( 1 + (-0.781 + 0.623i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.22432110185768280267521068671, −18.57870548005067467806441011563, −18.24597935853104736080945233344, −16.99965892518418468111238523660, −16.705268445431479440074255650240, −15.97470439333351682772947442511, −14.82817290720639131563620246570, −14.46027023774881847596333420726, −13.93401496015486213903035184007, −12.82770754092663641426883817824, −12.1333580474885023707620587350, −11.40420919663270922390017370438, −10.99615300871984486857098892660, −9.83114144647120127656275740224, −9.19045557975179427735617502196, −8.449788611107782098934346971541, −7.76802996868862534404156551853, −6.87992172764313103914132691621, −5.99599959321670604161249122490, −5.3562994632111019477384757775, −4.398593113012332070490528505978, −3.736831406737640526465366871207, −2.58332176336580934139887459566, −1.85160614495325736392758330159, −0.833515022917842278541104848532, 0.91929302185232810420059253909, 1.69819500297849965892907340254, 2.695320774141907264540183015824, 3.744915105325744961219415402131, 4.44305521462934124777973155897, 5.24123219547223591997077950909, 6.017027369183195439361796677441, 7.11716738762739467626924160881, 7.602372023014923393166563913588, 8.34535548983795022018067394832, 9.29312654143088477768205430564, 9.98294468072537345283670858569, 10.81635531082856436774685867975, 11.35469496114234446809804235073, 12.310405942330944147163016252875, 12.85655344446552750152563209598, 13.75880736397298248359401743506, 14.48047636444749454200520504184, 15.106995262972614915157693591988, 15.626644936689798476237901684806, 16.86005240112635390679398098228, 17.340087311905559926779607195557, 17.71790389827783236494305871614, 18.65765593749827027412241207285, 19.58762438497639275410193415836

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