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Properties

Label 1-4729-4729.135-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.102 - 0.994i$
Analytic cond. $21.9613$
Root an. cond. $21.9613$
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  + (−0.989 − 0.143i)2-s + (0.467 + 0.884i)3-s + (0.959 + 0.283i)4-s + (0.635 + 0.772i)5-s + (−0.336 − 0.941i)6-s + (0.0186 + 0.999i)7-s + (−0.908 − 0.417i)8-s + (−0.563 + 0.826i)9-s + (−0.518 − 0.855i)10-s + (0.987 + 0.158i)11-s + (0.198 + 0.980i)12-s + (−0.929 + 0.368i)13-s + (0.124 − 0.992i)14-s + (−0.385 + 0.922i)15-s + (0.839 + 0.543i)16-s + (−0.0981 − 0.995i)17-s + ⋯
L(s)  = 1  + (−0.989 − 0.143i)2-s + (0.467 + 0.884i)3-s + (0.959 + 0.283i)4-s + (0.635 + 0.772i)5-s + (−0.336 − 0.941i)6-s + (0.0186 + 0.999i)7-s + (−0.908 − 0.417i)8-s + (−0.563 + 0.826i)9-s + (−0.518 − 0.855i)10-s + (0.987 + 0.158i)11-s + (0.198 + 0.980i)12-s + (−0.929 + 0.368i)13-s + (0.124 − 0.992i)14-s + (−0.385 + 0.922i)15-s + (0.839 + 0.543i)16-s + (−0.0981 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.102 - 0.994i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.102 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1781758612 + 0.1974717662i\)
\(L(\frac12)\) \(\approx\) \(-0.1781758612 + 0.1974717662i\)
\(L(1)\) \(\approx\) \(0.6098232338 + 0.3903905084i\)
\(L(1)\) \(\approx\) \(0.6098232338 + 0.3903905084i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.989 - 0.143i)T \)
3 \( 1 + (0.467 + 0.884i)T \)
5 \( 1 + (0.635 + 0.772i)T \)
7 \( 1 + (0.0186 + 0.999i)T \)
11 \( 1 + (0.987 + 0.158i)T \)
13 \( 1 + (-0.929 + 0.368i)T \)
17 \( 1 + (-0.0981 - 0.995i)T \)
19 \( 1 + (-0.988 + 0.153i)T \)
23 \( 1 + (-0.663 - 0.748i)T \)
29 \( 1 + (-0.989 + 0.143i)T \)
31 \( 1 + (-0.366 - 0.930i)T \)
37 \( 1 + (-0.853 - 0.520i)T \)
41 \( 1 + (0.999 - 0.0318i)T \)
43 \( 1 + (-0.375 + 0.926i)T \)
47 \( 1 + (0.584 + 0.811i)T \)
53 \( 1 + (0.996 - 0.0849i)T \)
59 \( 1 + (0.756 + 0.653i)T \)
61 \( 1 + (-0.545 + 0.838i)T \)
67 \( 1 + (-0.295 - 0.955i)T \)
71 \( 1 + (-0.842 - 0.538i)T \)
73 \( 1 + (-0.997 - 0.0690i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.706 - 0.708i)T \)
89 \( 1 + (0.114 + 0.993i)T \)
97 \( 1 + (0.0186 + 0.999i)T \)
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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

−17.40953840625029294820040945488, −17.20342304622718214331506188615, −16.82177884069978107100430560463, −15.80754612043954046606469837109, −14.832231910483361115024036762202, −14.42265916763871145698942567383, −13.625383831693698379974092539302, −12.91984696626409682216245838260, −12.28907674929501779270982506724, −11.64885396487588404699370993886, −10.669282482075583341795171469916, −10.03502083939414378992631075179, −9.3629759821619860645305559182, −8.64055987173459638669912856218, −8.238564324762760656943488919807, −7.27970966793638558927594848369, −6.90802012065722507240077296432, −6.06160429439578382009177963096, −5.4607596070721661226043615574, −4.169622081244134735482444315764, −3.42854454337569829883552460248, −2.22299753176228496377836796365, −1.71655974664074941363864264930, −1.070669958787613927226382823919, −0.08669116286833041129754219578, 1.7418357390855011793784668886, 2.335285547734911806392672311722, 2.74514973031242711921244363406, 3.731392692219672094725612484490, 4.57845217687655233753517549375, 5.7051569966005210490985940352, 6.19025731212133261112385683564, 7.120388261188622034755197757861, 7.73367357493214475730659127053, 8.81558998126274607949915478066, 9.18542101203491507053201174823, 9.639788318167108094596983292057, 10.34669050115051926387038244291, 11.01913466436519251190449956166, 11.702122928261875442564867328500, 12.2881423032818840707193611704, 13.33934246703586269718120194442, 14.4040634535542939877604747421, 14.78155594529001760282140162546, 15.14182169389019548905525044873, 16.23219730235057791227056388149, 16.55374667626885534570825646934, 17.41908402911653125079481779673, 17.915656491742526463880288001, 18.81499792024323914323982826087

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