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Properties

Label 1-4560-4560.917-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.148 + 0.988i$
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 + (0.939 − 0.342i)13-s + (−0.642 − 0.766i)17-s + (−0.984 − 0.173i)23-s + (0.642 − 0.766i)29-s + (−0.5 + 0.866i)31-s − 37-s + (−0.939 − 0.342i)41-s + (0.173 + 0.984i)43-s + (0.642 − 0.766i)47-s + (0.5 + 0.866i)49-s + (−0.173 + 0.984i)53-s + (0.642 + 0.766i)59-s + (−0.984 − 0.173i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)11-s + (0.939 − 0.342i)13-s + (−0.642 − 0.766i)17-s + (−0.984 − 0.173i)23-s + (0.642 − 0.766i)29-s + (−0.5 + 0.866i)31-s − 37-s + (−0.939 − 0.342i)41-s + (0.173 + 0.984i)43-s + (0.642 − 0.766i)47-s + (0.5 + 0.866i)49-s + (−0.173 + 0.984i)53-s + (0.642 + 0.766i)59-s + (−0.984 − 0.173i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.148 + 0.988i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (917, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.148 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.092246732 + 0.9403175781i\)
\(L(\frac12)\) \(\approx\) \(1.092246732 + 0.9403175781i\)
\(L(1)\) \(\approx\) \(1.050021896 + 0.1519281923i\)
\(L(1)\) \(\approx\) \(1.050021896 + 0.1519281923i\)

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

−18.11164559508530412890061289454, −17.44403176721292869857411810160, −16.7855179670967401320617133926, −15.97296899090271544237853466828, −15.51645257557500587346360541793, −14.67471426077189256707890174777, −13.88605189377975671965615133151, −13.548341821629015594959822109025, −12.74168909303854366665067254222, −11.906282668600684840960751564005, −11.109744493964808160420285588243, −10.723982931668708793288223771529, −10.07543174415143016063918067582, −8.99044938556530202284253651333, −8.35601541849288399975730275822, −7.91203617105307046689116131384, −7.004857973722160604376991712039, −6.23758110843642631105401927641, −5.48706132740528323223008344788, −4.742129832123193275951076069047, −3.929805723995599131145847497871, −3.33245025597253954073210456575, −2.10001916483978757218729793640, −1.60484747514723937319298040660, −0.418842351445572286051096680304, 0.97582955730149965959587919154, 1.99253373074997582000736161062, 2.53745098585663583893773753194, 3.53089049904941452072157865516, 4.45387872131045874508514593444, 5.09405063809630451611714968829, 5.72991469019958704666992935809, 6.58830482156768490891223169958, 7.428571114233751318911274851595, 8.14706732333785990827001794237, 8.64629300525445248492260415030, 9.43030317542880266618392178114, 10.42415818967871149106907642728, 10.763564916165167631402552020888, 11.729987811956478753371466000797, 12.14140215353142683125897232985, 13.04009597644859716345020458823, 13.72183849763597528452818706495, 14.257972537341501185443302822425, 15.213464661726258519499211326151, 15.658998262048148654996116640581, 16.13964706922217306760758683000, 17.23478419679189997262529449763, 17.851829601872214453697622145369, 18.28943207556371037270143663658

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