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Properties

Label 1-4560-4560.3797-r0-0-0
Degree $1$
Conductor $4560$
Sign $-0.985 + 0.172i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.985 + 0.172i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (3797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ -0.985 + 0.172i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03537064934 + 0.4077996786i\)
\(L(\frac12)\) \(\approx\) \(0.03537064934 + 0.4077996786i\)
\(L(1)\) \(\approx\) \(0.9032106037 + 0.1169340822i\)
\(L(1)\) \(\approx\) \(0.9032106037 + 0.1169340822i\)

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

−17.97348450918140006278427090195, −17.07341337343665503475508261658, −16.68823015634859233564547489198, −16.06125632219215640401186213450, −14.964792143817376483046379818400, −14.633536065137192512412046715765, −13.960935858775692190263924734233, −13.18452133906327097788578975733, −12.56308653696416382998476551271, −11.68535807795031074646293311852, −11.14774204039136229819980065024, −10.431194769856194614042111228384, −9.82489769911498704441582419875, −8.937700232792890555260514020227, −8.14998701695421516267695820771, −7.61578227858102822123977267434, −6.97805997072046451744047288846, −5.98167814591841386158341184575, −5.21692844910309446283911852652, −4.65723510074263485559057247082, −3.81002503360698874277690400355, −2.94130010749604711673951600503, −2.07225354187149906212461022448, −1.28131202009190873998481879801, −0.105672720421360964032158454872, 1.31619513062054915576211894641, 2.05343115720687403897483042580, 2.870692841340934089061039842681, 3.63781871714514973600650778999, 4.74547538785302908033484184843, 5.37720877693806357653030219779, 5.630290459408702992786825700650, 7.06495624166696380228268557553, 7.50238101376881331613950054295, 8.15056625398292054507344804838, 8.91368811926033863070782319961, 9.76653775473617762553443123547, 10.32895308408706984244172562312, 11.09281541453203485896401212565, 11.8337045520925126027167813353, 12.44897628266700770601663183033, 13.03309915409222187866659020588, 13.941850059747264261243524322708, 14.58879960578617121780799254460, 15.24480718179346643311056889402, 15.6279447492811385087969371209, 16.66333423234280269674995552098, 17.22817005720712710658449973157, 18.05253146328660371402462543575, 18.28349535107901812763583826796

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