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Properties

Label 1-3840-3840.629-r1-0-0
Degree $1$
Conductor $3840$
Sign $0.219 - 0.975i$
Analytic cond. $412.665$
Root an. cond. $412.665$
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  + (0.195 − 0.980i)7-s + (0.773 − 0.634i)11-s + (−0.881 + 0.471i)13-s + (0.382 + 0.923i)17-s + (0.956 + 0.290i)19-s + (−0.831 − 0.555i)23-s + (−0.634 + 0.773i)29-s + (−0.707 − 0.707i)31-s + (0.290 + 0.956i)37-s + (0.555 − 0.831i)41-s + (0.995 + 0.0980i)43-s + (−0.923 + 0.382i)47-s + (−0.923 − 0.382i)49-s + (0.634 + 0.773i)53-s + (−0.881 − 0.471i)59-s + ⋯
L(s)  = 1  + (0.195 − 0.980i)7-s + (0.773 − 0.634i)11-s + (−0.881 + 0.471i)13-s + (0.382 + 0.923i)17-s + (0.956 + 0.290i)19-s + (−0.831 − 0.555i)23-s + (−0.634 + 0.773i)29-s + (−0.707 − 0.707i)31-s + (0.290 + 0.956i)37-s + (0.555 − 0.831i)41-s + (0.995 + 0.0980i)43-s + (−0.923 + 0.382i)47-s + (−0.923 − 0.382i)49-s + (0.634 + 0.773i)53-s + (−0.881 − 0.471i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.219 - 0.975i$
Analytic conductor: \(412.665\)
Root analytic conductor: \(412.665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3840,\ (1:\ ),\ 0.219 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.578251325 - 1.263146266i\)
\(L(\frac12)\) \(\approx\) \(1.578251325 - 1.263146266i\)
\(L(1)\) \(\approx\) \(1.080391940 - 0.1644428829i\)
\(L(1)\) \(\approx\) \(1.080391940 - 0.1644428829i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.195 - 0.980i)T \)
11 \( 1 + (0.773 - 0.634i)T \)
13 \( 1 + (-0.881 + 0.471i)T \)
17 \( 1 + (0.382 + 0.923i)T \)
19 \( 1 + (0.956 + 0.290i)T \)
23 \( 1 + (-0.831 - 0.555i)T \)
29 \( 1 + (-0.634 + 0.773i)T \)
31 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (0.290 + 0.956i)T \)
41 \( 1 + (0.555 - 0.831i)T \)
43 \( 1 + (0.995 + 0.0980i)T \)
47 \( 1 + (-0.923 + 0.382i)T \)
53 \( 1 + (0.634 + 0.773i)T \)
59 \( 1 + (-0.881 - 0.471i)T \)
61 \( 1 + (-0.0980 - 0.995i)T \)
67 \( 1 + (-0.0980 - 0.995i)T \)
71 \( 1 + (0.980 + 0.195i)T \)
73 \( 1 + (0.195 + 0.980i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (-0.290 + 0.956i)T \)
89 \( 1 + (0.831 - 0.555i)T \)
97 \( 1 + (-0.707 - 0.707i)T \)
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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

−18.32984279814124194526744451498, −17.91847169964204027804770392596, −17.36847735886187785332239124919, −16.31635716531747970704046645529, −15.92398571156474700070156516535, −14.898282671728182255311297407625, −14.68340345710905944376537432123, −13.81526168517122359447011629701, −12.97310993209917229729668228223, −12.06417013934166942935359311976, −11.9208074825904031921815685245, −11.08126098667647345694867104054, −9.99542829404608514468413750131, −9.42511876602625812649073898315, −9.02555265940232201233648587586, −7.77713348333618598513561724101, −7.499844098955218157871594587013, −6.52254256204340170839945071609, −5.589243021398376995595782855519, −5.16238114929918443689903964990, −4.25800709350764849998694203311, −3.309606559697022758252595008388, −2.49400458143807328868114480310, −1.80099813639563155788026832298, −0.72951466445856269217026965559, 0.387247903038007553333154838325, 1.27837305681817780707511421903, 2.02292775594900348577055405175, 3.2288134511637581295761786882, 3.87489747984198117675331460174, 4.50955972603534107013574134532, 5.485096630640431094722333253039, 6.23620785436918789908390185306, 7.01274218968161705136795620262, 7.71179946838279219021131053239, 8.309720644338372254632379013911, 9.39028626415369703093967967247, 9.773400162502505843007122807563, 10.762794612184779569228801549130, 11.197658809377256077460166319411, 12.12676485203715124625586277229, 12.63324271738148357347616592189, 13.62464160733007725958883454395, 14.26290797894573336471872268392, 14.53037027422919534324948678697, 15.53170318271296715942469338182, 16.555678249790762306617462405522, 16.74091233197042671705515415965, 17.39821401739968457810468874714, 18.300305587251757236329915143249

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