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Properties

Label 1-4729-4729.2585-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.825 - 0.564i$
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  + (−0.959 − 0.283i)2-s + (0.563 − 0.826i)3-s + (0.839 + 0.543i)4-s + (0.192 − 0.981i)5-s + (−0.773 + 0.633i)6-s + (0.999 − 0.0372i)7-s + (−0.651 − 0.758i)8-s + (−0.366 − 0.930i)9-s + (−0.462 + 0.886i)10-s + (−0.313 + 0.949i)11-s + (0.921 − 0.388i)12-s + (0.728 − 0.685i)13-s + (−0.968 − 0.247i)14-s + (−0.702 − 0.711i)15-s + (0.410 + 0.912i)16-s + (0.195 + 0.980i)17-s + ⋯
L(s)  = 1  + (−0.959 − 0.283i)2-s + (0.563 − 0.826i)3-s + (0.839 + 0.543i)4-s + (0.192 − 0.981i)5-s + (−0.773 + 0.633i)6-s + (0.999 − 0.0372i)7-s + (−0.651 − 0.758i)8-s + (−0.366 − 0.930i)9-s + (−0.462 + 0.886i)10-s + (−0.313 + 0.949i)11-s + (0.921 − 0.388i)12-s + (0.728 − 0.685i)13-s + (−0.968 − 0.247i)14-s + (−0.702 − 0.711i)15-s + (0.410 + 0.912i)16-s + (0.195 + 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.825 - 0.564i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (2585, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.825 - 0.564i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4926023681 - 1.592967816i\)
\(L(\frac12)\) \(\approx\) \(0.4926023681 - 1.592967816i\)
\(L(1)\) \(\approx\) \(0.8255751492 - 0.6044702196i\)
\(L(1)\) \(\approx\) \(0.8255751492 - 0.6044702196i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.959 - 0.283i)T \)
3 \( 1 + (0.563 - 0.826i)T \)
5 \( 1 + (0.192 - 0.981i)T \)
7 \( 1 + (0.999 - 0.0372i)T \)
11 \( 1 + (-0.313 + 0.949i)T \)
13 \( 1 + (0.728 - 0.685i)T \)
17 \( 1 + (0.195 + 0.980i)T \)
19 \( 1 + (0.952 - 0.303i)T \)
23 \( 1 + (0.992 + 0.119i)T \)
29 \( 1 + (-0.283 - 0.959i)T \)
31 \( 1 + (-0.681 - 0.732i)T \)
37 \( 1 + (0.458 + 0.888i)T \)
41 \( 1 + (-0.0637 - 0.997i)T \)
43 \( 1 + (-0.696 + 0.717i)T \)
47 \( 1 + (0.948 + 0.316i)T \)
53 \( 1 + (-0.169 - 0.985i)T \)
59 \( 1 + (-0.145 - 0.989i)T \)
61 \( 1 + (-0.914 + 0.405i)T \)
67 \( 1 + (-0.824 + 0.565i)T \)
71 \( 1 + (-0.419 - 0.907i)T \)
73 \( 1 + (-0.990 - 0.137i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (-0.00265 - 0.999i)T \)
89 \( 1 + (0.226 + 0.973i)T \)
97 \( 1 + (-0.999 + 0.0372i)T \)
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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

−18.42038463712886340487390586502, −18.03912889784807199295596436398, −16.908884950498427022993653443582, −16.46207465028883929619039149874, −15.79495566517131772252255891997, −15.18819771543995519227466710394, −14.407736234121592782621965470565, −14.12760231177387153340048667248, −13.487045974677525707820757681743, −11.91573339403652097446091359939, −11.18275698668708872366611658957, −10.94937773696313248238553316914, −10.3240519810985451569572323543, −9.42265880273948607026260794640, −8.904169912704157663771860330240, −8.33848839994183081139812051837, −7.37442915951741607952582634572, −7.14575959509423587411537754031, −5.81339940564612473241744015167, −5.468826167904605013230800912153, −4.47541952982559114953200774868, −3.24412968948560646667433546214, −2.967555724547988040958027177174, −1.946302606484806784350102105085, −1.15592681727297778031888752012, 0.58590561850825628139380939252, 1.44142854912556220102373737218, 1.756404824739796334185906486287, 2.68158635690481155311185782114, 3.58882253128315567025950794741, 4.50486256213504666526095347543, 5.53006613076318455215859520668, 6.18630555087867358100629384115, 7.3215031537601382904171503691, 7.72705031486830300005397535733, 8.28143647038550216586313023037, 8.8977302069123081222639294170, 9.52142567726372348257167197643, 10.290392334113580224154928012796, 11.18842715893923169611970483296, 11.82821312970827673097304747267, 12.44545062555246035734015900496, 13.17451322562263678318090858862, 13.48909822952583998340928321052, 14.8061523507660681353661011940, 15.14807310591299132829765626121, 15.932758361106131006076727914895, 16.93016762584459070202408706480, 17.486569173065441430487646939517, 17.775142393862124008148795881738

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