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Properties

Label 1-4928-4928.1691-r0-0-0
Degree $1$
Conductor $4928$
Sign $-0.476 + 0.879i$
Analytic cond. $22.8855$
Root an. cond. $22.8855$
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  + (−0.0261 + 0.999i)3-s + (0.824 + 0.566i)5-s + (−0.998 − 0.0523i)9-s + (−0.0784 − 0.996i)13-s + (−0.587 + 0.809i)15-s + (0.743 + 0.669i)17-s + (−0.477 − 0.878i)19-s + (−0.258 − 0.965i)23-s + (0.358 + 0.933i)25-s + (0.0784 − 0.996i)27-s + (−0.233 + 0.972i)29-s + (−0.978 + 0.207i)31-s + (0.999 − 0.0261i)37-s + (0.998 − 0.0523i)39-s + (0.156 − 0.987i)41-s + ⋯
L(s)  = 1  + (−0.0261 + 0.999i)3-s + (0.824 + 0.566i)5-s + (−0.998 − 0.0523i)9-s + (−0.0784 − 0.996i)13-s + (−0.587 + 0.809i)15-s + (0.743 + 0.669i)17-s + (−0.477 − 0.878i)19-s + (−0.258 − 0.965i)23-s + (0.358 + 0.933i)25-s + (0.0784 − 0.996i)27-s + (−0.233 + 0.972i)29-s + (−0.978 + 0.207i)31-s + (0.999 − 0.0261i)37-s + (0.998 − 0.0523i)39-s + (0.156 − 0.987i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4928\)    =    \(2^{6} \cdot 7 \cdot 11\)
Sign: $-0.476 + 0.879i$
Analytic conductor: \(22.8855\)
Root analytic conductor: \(22.8855\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4928} (1691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4928,\ (0:\ ),\ -0.476 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8777919910 + 1.474786766i\)
\(L(\frac12)\) \(\approx\) \(0.8777919910 + 1.474786766i\)
\(L(1)\) \(\approx\) \(1.031285046 + 0.4942461155i\)
\(L(1)\) \(\approx\) \(1.031285046 + 0.4942461155i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.0261 + 0.999i)T \)
5 \( 1 + (0.824 + 0.566i)T \)
13 \( 1 + (-0.0784 - 0.996i)T \)
17 \( 1 + (0.743 + 0.669i)T \)
19 \( 1 + (-0.477 - 0.878i)T \)
23 \( 1 + (-0.258 - 0.965i)T \)
29 \( 1 + (-0.233 + 0.972i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.999 - 0.0261i)T \)
41 \( 1 + (0.156 - 0.987i)T \)
43 \( 1 + (-0.382 + 0.923i)T \)
47 \( 1 + (0.994 - 0.104i)T \)
53 \( 1 + (-0.182 + 0.983i)T \)
59 \( 1 + (-0.477 + 0.878i)T \)
61 \( 1 + (-0.182 - 0.983i)T \)
67 \( 1 + (0.608 - 0.793i)T \)
71 \( 1 + (-0.453 + 0.891i)T \)
73 \( 1 + (0.629 + 0.777i)T \)
79 \( 1 + (-0.743 + 0.669i)T \)
83 \( 1 + (0.649 + 0.760i)T \)
89 \( 1 + (-0.965 + 0.258i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−17.94554294057638202116780121144, −17.12974952038716637268085303653, −16.71872624033966453771039653514, −16.16496958059097112713172843468, −14.98494263156777190319266867546, −14.27588199981427989959985900909, −13.823724437698829316547786141725, −13.16772842411155262095544180882, −12.60192279209612607092114171679, −11.82074699672511569852510236507, −11.43892696201203071057029611406, −10.33789228682879503636912675258, −9.549411563670388790175179885124, −9.09911277280136451510185185042, −8.19667236148029200871248634835, −7.60427183161976845007347293500, −6.853576771902303848534728882928, −5.98558665139085538354157470647, −5.682726990051852778003028498160, −4.75162885535970268883330835942, −3.83080400907691973942092783263, −2.82470183664777092502577240396, −1.908252885647375779870666203009, −1.58637486716394431611525621056, −0.4932338175288653386415910473, 0.95074589799804398634474061431, 2.18688427423296347805853024240, 2.84104290904813738699028431703, 3.50966418979621328904636958829, 4.357960451532645476097964979436, 5.242393636596294573428745954978, 5.73718069727220309390996346659, 6.4178717921117172058209452586, 7.31793062696813119703426159662, 8.208785946204418513350176418355, 8.93993007982815833802714691542, 9.57344585744212918910186669949, 10.28319903739168358900236730423, 10.766925555627123530117387843942, 11.17783566793222416669242181882, 12.43675002599183032687411536844, 12.84044688396135093508068724874, 13.858110371577224785019035046470, 14.40780788839777263436400158860, 14.99681973710251440726610081840, 15.4629517348666316833420003397, 16.41831547322752504442532106224, 16.942586270559919430414856285446, 17.52894514022142441526258223415, 18.21792075714710857218638365991

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