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Properties

Label 1-4560-4560.2387-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.661 + 0.750i$
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  + i·7-s − i·11-s + (0.5 + 0.866i)13-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)23-s + (−0.866 + 0.5i)29-s + 31-s − 37-s + (0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (0.866 − 0.5i)47-s − 49-s + (0.5 + 0.866i)53-s + (0.866 + 0.5i)59-s + (0.866 − 0.5i)61-s + ⋯
L(s)  = 1  + i·7-s − i·11-s + (0.5 + 0.866i)13-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)23-s + (−0.866 + 0.5i)29-s + 31-s − 37-s + (0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (0.866 − 0.5i)47-s − 49-s + (0.5 + 0.866i)53-s + (0.866 + 0.5i)59-s + (0.866 − 0.5i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.661 + 0.750i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (2387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.661 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.739393488 + 0.7855244945i\)
\(L(\frac12)\) \(\approx\) \(1.739393488 + 0.7855244945i\)
\(L(1)\) \(\approx\) \(1.153023646 + 0.1790721955i\)
\(L(1)\) \(\approx\) \(1.153023646 + 0.1790721955i\)

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 \( 1 + iT \)
11 \( 1 - iT \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−17.99067987931230753340216745770, −17.43741374601666079380998915226, −16.92799435430608073150315547313, −16.13638693569587491835353197829, −15.40654077967479377307636817588, −14.842090510305069212188179172333, −14.05251593340354385695434834666, −13.373015187975213053002199430379, −12.86700273431825418258114798391, −12.04193211880963019435024347786, −11.33112826715584371132733917310, −10.5382096013669865690176763289, −10.00599208708201898254482709687, −9.41346904896389742666480369799, −8.40507284860222901016398581071, −7.66409964075960375032807618854, −7.1953755467803275123051976538, −6.439335555033208307544953581612, −5.42544132781288211853195200344, −4.89277001036948972766995855201, −3.92757862205523121924418213016, −3.37852974666671753917796840225, −2.43090393728681276971255966669, −1.38784041765844358679727567998, −0.66256893145025133420222643227, 0.900779671231877332218015708141, 1.77911313830513516671316504329, 2.687355721239165969725122025315, 3.40237268889172534456967802482, 4.16932773134595241417827773142, 5.22078662169645612008251004473, 5.74097645078262351983404483716, 6.42876920526703887325507340063, 7.20251552507650505023844169171, 8.20990435799636111075470787293, 8.756916892485248269593427253567, 9.19770412784605141926881783488, 10.2069309852860121679792909554, 10.88384591829076204071619835074, 11.63554174027645670975893497254, 12.10942464274861910756030838807, 12.944574501669432803528772319407, 13.58021257897984477293395960594, 14.40839460264218303670996234996, 14.85385513461589707268533645950, 15.76622104789025392326671359924, 16.24494115666466860316948273921, 16.91989400892366645300061197502, 17.62255797669585656744881443341, 18.63119449765247218223231125158

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