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Properties

Label 1-5328-5328.2981-r1-0-0
Degree $1$
Conductor $5328$
Sign $-0.951 - 0.308i$
Analytic cond. $572.573$
Root an. cond. $572.573$
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  + (0.642 + 0.766i)5-s + (−0.173 − 0.984i)7-s − i·11-s + (0.342 − 0.939i)13-s + (0.173 − 0.984i)17-s + (0.642 + 0.766i)19-s − 23-s + (−0.173 + 0.984i)25-s − i·29-s + (0.5 − 0.866i)31-s + (0.642 − 0.766i)35-s + (−0.939 − 0.342i)41-s + (−0.866 + 0.5i)43-s + (0.5 − 0.866i)47-s + (−0.939 + 0.342i)49-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)5-s + (−0.173 − 0.984i)7-s − i·11-s + (0.342 − 0.939i)13-s + (0.173 − 0.984i)17-s + (0.642 + 0.766i)19-s − 23-s + (−0.173 + 0.984i)25-s − i·29-s + (0.5 − 0.866i)31-s + (0.642 − 0.766i)35-s + (−0.939 − 0.342i)41-s + (−0.866 + 0.5i)43-s + (0.5 − 0.866i)47-s + (−0.939 + 0.342i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5328\)    =    \(2^{4} \cdot 3^{2} \cdot 37\)
Sign: $-0.951 - 0.308i$
Analytic conductor: \(572.573\)
Root analytic conductor: \(572.573\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5328} (2981, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5328,\ (1:\ ),\ -0.951 - 0.308i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2616224497 - 1.654207883i\)
\(L(\frac12)\) \(\approx\) \(0.2616224497 - 1.654207883i\)
\(L(1)\) \(\approx\) \(1.091645786 - 0.2779807699i\)
\(L(1)\) \(\approx\) \(1.091645786 - 0.2779807699i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( 1 \)
good5 \( 1 + (0.642 + 0.766i)T \)
7 \( 1 + (-0.173 - 0.984i)T \)
11 \( 1 - iT \)
13 \( 1 + (0.342 - 0.939i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 - T \)
29 \( 1 - iT \)
31 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.342 + 0.939i)T \)
59 \( 1 + (0.984 + 0.173i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (-0.342 - 0.939i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−17.96243168390658579273774227451, −17.51596712296115378134920200353, −16.743104758624747642763951375480, −15.99994678939724050790587386323, −15.6159322747252428017900882525, −14.668216694111843976897753394952, −14.16517799318109211297972968160, −13.264681989478803738905700416959, −12.78398322692897897650989997894, −12.03223736712319021598758522466, −11.70318965619538638524006514043, −10.55205200134656663628145576817, −9.83413079253891854986994470764, −9.35809660627809105318713949052, −8.605054725982554717770563138250, −8.20501451954875497212500278382, −6.93988590062447299323703729479, −6.50854381391673862096008577822, −5.58673197889493784529884279736, −5.08436280203103084702238177966, −4.34673047519155327810942049617, −3.46336654373446033872686029285, −2.39070913609503657410235661085, −1.82488098120203731444102610530, −1.15806600208644625410694614307, 0.2457017456745629188017659586, 0.89639392987987948725540323157, 1.90057618843921755228914336747, 2.89981156432444488685870389163, 3.394320890458537653007793529738, 4.11390896827139318483998003064, 5.23661709925071224749259492819, 5.91895022608745276336304210708, 6.379111494582638301052641563628, 7.37372923862408891211592683505, 7.78420734533388811538281590159, 8.62861473905198494260466865016, 9.64668855859353733706162815741, 10.19049109817083812928448955063, 10.52719820519509446845288912744, 11.515863905251060858104330543399, 11.88515389713549276276988403503, 13.24867948843912350363902743770, 13.48339927185866750170646476955, 14.01897420215665289943874706044, 14.69033324808491501299734845103, 15.545669848104321525211901269337, 16.20131060051351601282992394575, 16.85173911191161219631247142890, 17.45313511246143368138248453370

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