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Properties

Label 1-4176-4176.2419-r0-0-0
Degree $1$
Conductor $4176$
Sign $-0.397 + 0.917i$
Analytic cond. $19.3932$
Root an. cond. $19.3932$
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)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + i·17-s + 19-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s + (0.866 + 0.5i)31-s − i·35-s − 37-s + (−0.866 − 0.5i)41-s + (0.5 + 0.866i)43-s + (0.866 − 0.5i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + i·17-s + 19-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s + (0.866 + 0.5i)31-s − i·35-s − 37-s + (−0.866 − 0.5i)41-s + (0.5 + 0.866i)43-s + (0.866 − 0.5i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4176\)    =    \(2^{4} \cdot 3^{2} \cdot 29\)
Sign: $-0.397 + 0.917i$
Analytic conductor: \(19.3932\)
Root analytic conductor: \(19.3932\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4176} (2419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4176,\ (0:\ ),\ -0.397 + 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4188427607 + 0.6376357103i\)
\(L(\frac12)\) \(\approx\) \(0.4188427607 + 0.6376357103i\)
\(L(1)\) \(\approx\) \(0.8420859207 + 0.04388116637i\)
\(L(1)\) \(\approx\) \(0.8420859207 + 0.04388116637i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
29 \( 1 \)
good5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + iT \)
19 \( 1 + T \)
23 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 - iT \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−18.406200214255183141986609472054, −17.68620546642207231817826088208, −16.57147730118371928009329259253, −16.03816626660248665149036641849, −15.58636162782782130118498291889, −14.97235269684989573754089177935, −13.93175836584794471893361722354, −13.69343487416838007501926792943, −12.52334120078925148616912862907, −11.93798690089555375305052294321, −11.489851522470770577828177971539, −10.70287057663900630259181666882, −9.97534805448596374716166943903, −8.9966830841803756350795197718, −8.57445067529636563166723293624, −7.76849862497899239275042450766, −6.99317309437009679005277172039, −6.1940412724943671913407715241, −5.69079631364499222400312147230, −4.67208587203120164900246767612, −3.76176507199308880828745344217, −3.08002355498186406292703938297, −2.63329933192057322149501903262, −1.23267916519244336658834882357, −0.2423228017727836044672820687, 1.165541419533632074923224944666, 1.610057062660765933170370416406, 3.09291907939441940123320169780, 3.85173335388771709066474348294, 4.166531236014313624902541895737, 5.09803592257825450451076088689, 6.04992237412541854652592441505, 6.91855818786181425346335463414, 7.3658116052571243652981694421, 8.22295172170090211632261175651, 8.86961441072012863077776132656, 9.701837684494088793342606548, 10.30811704926341265123080666770, 11.162881421912296090206072317036, 11.85700015040022518141174114025, 12.38315727664626993238741082815, 13.19766233081531652976333716198, 13.790076842197508015105737909987, 14.50908009239335888892528899850, 15.503762911697344247877618948804, 15.817643639460819245454488060406, 16.56210723626227779829294425368, 17.22030581210440857665912345355, 17.7765074977027110098647878990, 18.7737239781798425963764635212

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