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Properties

Label 1-4560-4560.3659-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.646 + 0.762i$
Analytic cond. $21.1765$
Root an. cond. $21.1765$
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 − i·11-s + (−0.866 − 0.5i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)23-s + (−0.866 − 0.5i)29-s − 31-s − i·37-s + (−0.5 − 0.866i)41-s + (0.866 − 0.5i)43-s + (0.5 − 0.866i)47-s + 49-s + (−0.866 − 0.5i)53-s + (0.866 − 0.5i)59-s + (0.866 + 0.5i)61-s + ⋯
L(s)  = 1  − 7-s − i·11-s + (−0.866 − 0.5i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)23-s + (−0.866 − 0.5i)29-s − 31-s − i·37-s + (−0.5 − 0.866i)41-s + (0.866 − 0.5i)43-s + (0.5 − 0.866i)47-s + 49-s + (−0.866 − 0.5i)53-s + (0.866 − 0.5i)59-s + (0.866 + 0.5i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.646 + 0.762i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (3659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.646 + 0.762i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7351190635 + 0.3404171429i\)
\(L(\frac12)\) \(\approx\) \(0.7351190635 + 0.3404171429i\)
\(L(1)\) \(\approx\) \(0.7841658291 + 0.01966728395i\)
\(L(1)\) \(\approx\) \(0.7841658291 + 0.01966728395i\)

Euler product

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

−18.327746199862214778130906023881, −17.251176612694334175710267645604, −16.77561971507051050313728834822, −16.19670537798920758187061384293, −15.57019588991883180232131957618, −14.65286015724764183677299891678, −14.21771545966551397097849896968, −13.24974867405724310419863344133, −12.82974835524378403176438188804, −12.14004890991783506821546451147, −11.27823841127171005312060656091, −10.67213015999231682444050001687, −9.89842470752214475468549126179, −9.24427383513730599386732340420, −8.611483064025219178868523216221, −7.8037229031195408336132429683, −6.94910326177536639210040831326, −6.29740173207557012110799698920, −5.76099386301656971581857394583, −4.7527876183931145208641602943, −3.96485581331869599693469561438, −3.23830178660766078351079522940, −2.49116761485323079290379515212, −1.57771224028642565216506265016, −0.33660062116835989489163525036, 0.609371010617851225997681473648, 2.086468935115898486680312969706, 2.43683467030864270282666073265, 3.59252618515000407761803037494, 4.071280247318712937396789579364, 5.25209201468266643640263323125, 5.56460691436320106885384587674, 6.76377780956783637012506775220, 7.18145247315821367454567084718, 7.79365962280247026552404985803, 8.929771236416925899815042134667, 9.56871180561051040467827165190, 9.92239538375384396873676098768, 10.788829673067474093765392779685, 11.63029909523301968552484202131, 12.36999773795428077635429251587, 12.833002832336708279193084519043, 13.52687650163027225125284644600, 14.305722845160256209414639427759, 15.07473067697769350932819651937, 15.656300653953985560130054874730, 16.202396334418991383109991848297, 17.08049474042440298103982888188, 17.60501980172264496514375256052, 18.26010074552224279570827774659

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