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Properties

Label 1-2580-2580.2183-r0-0-0
Degree $1$
Conductor $2580$
Sign $0.704 + 0.709i$
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

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2580\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 43\)
Sign: $0.704 + 0.709i$
Analytic conductor: \(11.9814\)
Root analytic conductor: \(11.9814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2580} (2183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2580,\ (0:\ ),\ 0.704 + 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.010102203 + 0.8367752523i\)
\(L(\frac12)\) \(\approx\) \(2.010102203 + 0.8367752523i\)
\(L(1)\) \(\approx\) \(1.313727993 + 0.2091392586i\)
\(L(1)\) \(\approx\) \(1.313727993 + 0.2091392586i\)

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 \( 1 + (0.680 - 0.733i)T \)
17 \( 1 + (-0.294 + 0.955i)T \)
19 \( 1 + (0.988 + 0.149i)T \)
23 \( 1 + (0.930 + 0.365i)T \)
29 \( 1 + (-0.0747 + 0.997i)T \)
31 \( 1 + (-0.826 - 0.563i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (-0.781 - 0.623i)T \)
53 \( 1 + (0.680 + 0.733i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (-0.826 + 0.563i)T \)
67 \( 1 + (0.149 - 0.988i)T \)
71 \( 1 + (-0.365 - 0.930i)T \)
73 \( 1 + (-0.680 + 0.733i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.997 + 0.0747i)T \)
89 \( 1 + (-0.0747 - 0.997i)T \)
97 \( 1 + (0.781 - 0.623i)T \)
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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

−19.23669092582001574113985090213, −18.56425364418257441587043554063, −17.8985480576160042995312484188, −17.196673967164911191357996082333, −16.35268472752712635003956002969, −16.01134939691344376557742583258, −14.83015794836685663968442096001, −14.29023102776892631660644805169, −13.655449110026992553224697388002, −13.06940363016653527257614179006, −11.80015744216720671743075811911, −11.36591591302409768207343314254, −10.92698249819878381558713729278, −9.76410467471095039143826981556, −9.0943454063275304007773072552, −8.40091973023544602634517682675, −7.54317764978320925841173932709, −6.84086754200049227333184668433, −6.032250792855195404917851706658, −5.073235515157142852900610379045, −4.38929850559350701460516947830, −3.55848612602915059312779695487, −2.65640341184438631338986817755, −1.48702352004117082081539847172, −0.82165229891564954769907912193, 1.15543113998124375177598973324, 1.76269669219217405386000066219, 2.833196680082392100593865422400, 3.77328114673825484866457525169, 4.55785600386135147336826802418, 5.48383352259565781330702487690, 5.992027108500829248927903630687, 7.176405134710799141562638278242, 7.68294394320996762636691891042, 8.68435053116529019560250122207, 9.12858774477627727530057488023, 10.108203575449591510632583996264, 10.96946166124774567676590843543, 11.45048737586070158978576075537, 12.38308502864197605194453119561, 12.91484016932467863898565558412, 13.81033416551769492046484721080, 14.75035096719633693202879299278, 15.01416237442469241753526370124, 15.8281114414341960914425628691, 16.749448375384540174010043857835, 17.434149098329543464947624621118, 18.15455493836660205915140222744, 18.46902345871633171233247692132, 19.75241705779476041762836858807

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