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Properties

Label 1-3920-3920.53-r1-0-0
Degree $1$
Conductor $3920$
Sign $-0.916 - 0.399i$
Analytic cond. $421.262$
Root an. cond. $421.262$
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  + (0.0747 + 0.997i)3-s + (−0.988 + 0.149i)9-s + (0.149 − 0.988i)11-s + (0.623 − 0.781i)13-s + (−0.294 − 0.955i)17-s + (−0.866 − 0.5i)19-s + (0.294 − 0.955i)23-s + (−0.222 − 0.974i)27-s + (−0.974 − 0.222i)29-s + (−0.5 − 0.866i)31-s + (0.997 + 0.0747i)33-s + (−0.733 − 0.680i)37-s + (0.826 + 0.563i)39-s + (0.900 + 0.433i)41-s + (0.900 − 0.433i)43-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)3-s + (−0.988 + 0.149i)9-s + (0.149 − 0.988i)11-s + (0.623 − 0.781i)13-s + (−0.294 − 0.955i)17-s + (−0.866 − 0.5i)19-s + (0.294 − 0.955i)23-s + (−0.222 − 0.974i)27-s + (−0.974 − 0.222i)29-s + (−0.5 − 0.866i)31-s + (0.997 + 0.0747i)33-s + (−0.733 − 0.680i)37-s + (0.826 + 0.563i)39-s + (0.900 + 0.433i)41-s + (0.900 − 0.433i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.916 - 0.399i$
Analytic conductor: \(421.262\)
Root analytic conductor: \(421.262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3920,\ (1:\ ),\ -0.916 - 0.399i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1894823277 - 0.9094734455i\)
\(L(\frac12)\) \(\approx\) \(0.1894823277 - 0.9094734455i\)
\(L(1)\) \(\approx\) \(0.9517360618 + 0.01144377788i\)
\(L(1)\) \(\approx\) \(0.9517360618 + 0.01144377788i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.0747 + 0.997i)T \)
11 \( 1 + (0.149 - 0.988i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (-0.294 - 0.955i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.294 - 0.955i)T \)
29 \( 1 + (-0.974 - 0.222i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.733 - 0.680i)T \)
41 \( 1 + (0.900 + 0.433i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.930 - 0.365i)T \)
53 \( 1 + (0.733 - 0.680i)T \)
59 \( 1 + (0.563 - 0.826i)T \)
61 \( 1 + (-0.680 + 0.733i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.930 - 0.365i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
97 \( 1 - iT \)
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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

−18.67774201617672058392442059562, −18.00381595765330199906118254371, −17.28300097502762808919967315792, −16.9203080646102091733087674711, −15.909999225865593595346346227303, −15.07311379242810524390143844832, −14.540910723493421057957809020047, −13.77813315880541601097928552631, −13.13130049221659871966509679560, −12.446153406314004240189999066575, −12.027929284235902987177033706764, −11.01891824362250888084059006008, −10.59965641771231975895037246107, −9.339125155871060331292665528976, −8.942570911672248259395758415555, −8.0685685960943936075950331229, −7.368468921942488469809401681571, −6.74180595500816163212577430455, −6.07511391179365745494412054667, −5.3393987313346894000977047728, −4.21153653937738348581042134968, −3.633339984051722285334543422181, −2.480756130436569213648624003687, −1.72150970860589305439413824279, −1.27572985807555068582017304141, 0.16872308590715334891400935280, 0.74087848658642099242845580877, 2.2950246375293029403879327058, 2.87889931299801815533630125991, 3.82453756866096616418500698674, 4.27899748762625616842193724171, 5.39546587061077409590644230584, 5.743719927628259211015698312423, 6.68262145546835391908594878423, 7.64777692955694842754155249496, 8.57684415233828175656099679249, 8.90858373898834181053090646677, 9.68348212260580017126128778616, 10.630919838174955022923077775471, 10.99267759679670460260489854334, 11.57238248992171091658450528785, 12.64515262448842193998048105028, 13.36827463126375353891190277042, 14.01418812371291877733285417048, 14.789156807622990966408390396113, 15.33255483262134238245490393176, 16.10810513254367652785361328744, 16.49868296440758578536766272989, 17.28277117060748284504781373534, 17.96258753605067793406497545307

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