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Properties

Label 1-4560-4560.557-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.286 + 0.958i$
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  + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)11-s + (0.766 − 0.642i)13-s + (−0.984 − 0.173i)17-s + (−0.342 + 0.939i)23-s + (−0.984 + 0.173i)29-s + (−0.5 − 0.866i)31-s + 37-s + (0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (0.984 − 0.173i)47-s + (0.5 − 0.866i)49-s + (−0.939 − 0.342i)53-s + (−0.984 − 0.173i)59-s + (0.342 − 0.939i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)11-s + (0.766 − 0.642i)13-s + (−0.984 − 0.173i)17-s + (−0.342 + 0.939i)23-s + (−0.984 + 0.173i)29-s + (−0.5 − 0.866i)31-s + 37-s + (0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (0.984 − 0.173i)47-s + (0.5 − 0.866i)49-s + (−0.939 − 0.342i)53-s + (−0.984 − 0.173i)59-s + (0.342 − 0.939i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.286 + 0.958i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.286 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6597491146 + 0.4913076933i\)
\(L(\frac12)\) \(\approx\) \(0.6597491146 + 0.4913076933i\)
\(L(1)\) \(\approx\) \(0.8182403873 + 0.03249233461i\)
\(L(1)\) \(\approx\) \(0.8182403873 + 0.03249233461i\)

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 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (-0.984 + 0.173i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.984 - 0.173i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.984 - 0.173i)T \)
61 \( 1 + (0.342 - 0.939i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.642 + 0.766i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.984 + 0.173i)T \)
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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

−18.10784203657184145799272087533, −17.47217708206872183839855671248, −16.615508663306208467031989198424, −16.053938089077177889233233537557, −15.61540822521194165762434217718, −14.713940055987181469196505140139, −13.989347166428329769111964699952, −13.24866604958960032716470177298, −12.83651662492662182838047295004, −12.14313033848507872535788300908, −11.02855545725303532115280840075, −10.73360374322564612702392217048, −9.91856107333968805909664875652, −9.15311269405299509916558307609, −8.62561429059808502812074235236, −7.5354414556209350425064247443, −7.12392482838167749911973393815, −6.192217122336570190956657625836, −5.757466686085003881350501983603, −4.39917245441084179030146895393, −4.24232294561592464485120363840, −3.122409297205967448791157054932, −2.41163871771453597298735680112, −1.51180820865849602666854249414, −0.29692998087968488828110492701, 0.7334921659965644305976673386, 1.996641745156274569922702210838, 2.73391584158453108156377411757, 3.43958763344735576617496724177, 4.18250866425474918388028805361, 5.28041357452933365600281373953, 5.88358323428689741447327450380, 6.34985705744295321103138807054, 7.47502386499665945893756625865, 7.94015687850282442743934612888, 8.936333319839233606816711264445, 9.34691350291050731207041908187, 10.18427571489599091926942415638, 11.0262733676352057694430231088, 11.37418516467665174474325985469, 12.46658417277075278978177612587, 13.13871640667611839406002182962, 13.3355645818726529211475531630, 14.3044814352573323898139417584, 15.292870377819338050890681895787, 15.67313551733412903977465853471, 16.17208783299717291969464172622, 16.987487722074652730303457562364, 17.828700048991614882709906980705, 18.43696073148642020090578790665

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