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Properties

Label 1-4024-4024.1187-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.754 - 0.656i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
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  + (0.580 + 0.814i)3-s + (−0.803 + 0.595i)5-s + (0.0812 − 0.996i)7-s + (−0.325 + 0.945i)9-s + (−0.106 − 0.994i)11-s + (0.817 − 0.575i)13-s + (−0.951 − 0.307i)15-s + (−0.0937 − 0.995i)17-s + (0.883 − 0.468i)19-s + (0.858 − 0.512i)21-s + (−0.930 + 0.366i)23-s + (0.289 − 0.957i)25-s + (−0.958 + 0.283i)27-s + (−0.630 − 0.776i)29-s + (−0.590 + 0.806i)31-s + ⋯
L(s)  = 1  + (0.580 + 0.814i)3-s + (−0.803 + 0.595i)5-s + (0.0812 − 0.996i)7-s + (−0.325 + 0.945i)9-s + (−0.106 − 0.994i)11-s + (0.817 − 0.575i)13-s + (−0.951 − 0.307i)15-s + (−0.0937 − 0.995i)17-s + (0.883 − 0.468i)19-s + (0.858 − 0.512i)21-s + (−0.930 + 0.366i)23-s + (0.289 − 0.957i)25-s + (−0.958 + 0.283i)27-s + (−0.630 − 0.776i)29-s + (−0.590 + 0.806i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.754 - 0.656i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.754 - 0.656i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1377924865 - 0.3683257449i\)
\(L(\frac12)\) \(\approx\) \(0.1377924865 - 0.3683257449i\)
\(L(1)\) \(\approx\) \(0.9319770905 + 0.08692802468i\)
\(L(1)\) \(\approx\) \(0.9319770905 + 0.08692802468i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.580 + 0.814i)T \)
5 \( 1 + (-0.803 + 0.595i)T \)
7 \( 1 + (0.0812 - 0.996i)T \)
11 \( 1 + (-0.106 - 0.994i)T \)
13 \( 1 + (0.817 - 0.575i)T \)
17 \( 1 + (-0.0937 - 0.995i)T \)
19 \( 1 + (0.883 - 0.468i)T \)
23 \( 1 + (-0.930 + 0.366i)T \)
29 \( 1 + (-0.630 - 0.776i)T \)
31 \( 1 + (-0.590 + 0.806i)T \)
37 \( 1 + (0.360 - 0.932i)T \)
41 \( 1 + (0.704 + 0.709i)T \)
43 \( 1 + (-0.277 + 0.960i)T \)
47 \( 1 + (-0.998 - 0.0500i)T \)
53 \( 1 + (-0.883 - 0.468i)T \)
59 \( 1 + (0.864 - 0.501i)T \)
61 \( 1 + (-0.747 - 0.663i)T \)
67 \( 1 + (-0.951 + 0.307i)T \)
71 \( 1 + (-0.668 + 0.743i)T \)
73 \( 1 + (0.474 + 0.880i)T \)
79 \( 1 + (-0.997 - 0.0750i)T \)
83 \( 1 + (0.118 - 0.992i)T \)
89 \( 1 + (0.180 + 0.983i)T \)
97 \( 1 + (-0.739 + 0.673i)T \)
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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

−18.579226417861264279981953882466, −18.36364827498262933758315657422, −17.54879669549231940387508026766, −16.614244347657308085084187287800, −15.95902700601935146024101063552, −15.15860861049238095715035051266, −14.82608754803857312929052104246, −13.95428825208934950677764099464, −13.05402064540239810009534215186, −12.62003026745104925838180562871, −11.92828441167403897961874886518, −11.58825748839895947420982536615, −10.474317225886713355568228634249, −9.32988472609291723967325090976, −9.02267835013547292275305073182, −8.14058648107658484204933161993, −7.79836800310061310975737025012, −6.923874674327764760329121587246, −6.086217294802354311584159233144, −5.42016055442503895515964703537, −4.32130940549206839380413891295, −3.73190148320045639086656074631, −2.84551549776959298400964226968, −1.73920688295225635976883853664, −1.51169220026784047096309070380, 0.10056601966025552878658681599, 1.21582641150691929763501996643, 2.613067574258417911378983877084, 3.30785747008158253917196877051, 3.70751178766302452007898830373, 4.48479891345161435845311270226, 5.333480832220370741974960554264, 6.226861166249883854970702131683, 7.22955690083029454078643613590, 7.85623216477302824386275785827, 8.26985795355336924165218133251, 9.296004138912842083683059687056, 9.907606803510613947390834182049, 10.74948925437324028626417216622, 11.21499381148832973331896547423, 11.61508591179090024284315451050, 13.04831185041768783442017469400, 13.55891043653160881933376539570, 14.26925092156886524892680350439, 14.65497101860449534027991152539, 15.78163943859269972431104703725, 16.03052840837565047292352380202, 16.4370594474037339120563513120, 17.64764925330202241153360501820, 18.202662169677167093157472375

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