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Properties

Label 1-4024-4024.1363-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.651 + 0.758i$
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  + (−0.668 + 0.743i)3-s + (−0.925 + 0.378i)5-s + (0.372 + 0.928i)7-s + (−0.106 − 0.994i)9-s + (−0.831 + 0.554i)11-s + (0.549 − 0.835i)13-s + (0.337 − 0.941i)15-s + (0.630 − 0.776i)17-s + (−0.659 + 0.752i)19-s + (−0.939 − 0.343i)21-s + (0.915 − 0.401i)23-s + (0.713 − 0.700i)25-s + (0.810 + 0.585i)27-s + (−0.787 − 0.615i)29-s + (−0.858 − 0.512i)31-s + ⋯
L(s)  = 1  + (−0.668 + 0.743i)3-s + (−0.925 + 0.378i)5-s + (0.372 + 0.928i)7-s + (−0.106 − 0.994i)9-s + (−0.831 + 0.554i)11-s + (0.549 − 0.835i)13-s + (0.337 − 0.941i)15-s + (0.630 − 0.776i)17-s + (−0.659 + 0.752i)19-s + (−0.939 − 0.343i)21-s + (0.915 − 0.401i)23-s + (0.713 − 0.700i)25-s + (0.810 + 0.585i)27-s + (−0.787 − 0.615i)29-s + (−0.858 − 0.512i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.651 + 0.758i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.651 + 0.758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3092421526 + 0.6734280906i\)
\(L(\frac12)\) \(\approx\) \(0.3092421526 + 0.6734280906i\)
\(L(1)\) \(\approx\) \(0.6429741077 + 0.2675019043i\)
\(L(1)\) \(\approx\) \(0.6429741077 + 0.2675019043i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.668 + 0.743i)T \)
5 \( 1 + (-0.925 + 0.378i)T \)
7 \( 1 + (0.372 + 0.928i)T \)
11 \( 1 + (-0.831 + 0.554i)T \)
13 \( 1 + (0.549 - 0.835i)T \)
17 \( 1 + (0.630 - 0.776i)T \)
19 \( 1 + (-0.659 + 0.752i)T \)
23 \( 1 + (0.915 - 0.401i)T \)
29 \( 1 + (-0.787 - 0.615i)T \)
31 \( 1 + (-0.858 - 0.512i)T \)
37 \( 1 + (-0.845 - 0.533i)T \)
41 \( 1 + (-0.997 + 0.0750i)T \)
43 \( 1 + (0.889 - 0.457i)T \)
47 \( 1 + (-0.360 + 0.932i)T \)
53 \( 1 + (0.659 + 0.752i)T \)
59 \( 1 + (0.998 + 0.0500i)T \)
61 \( 1 + (-0.143 - 0.989i)T \)
67 \( 1 + (0.337 + 0.941i)T \)
71 \( 1 + (0.289 + 0.957i)T \)
73 \( 1 + (0.764 - 0.644i)T \)
79 \( 1 + (0.229 + 0.973i)T \)
83 \( 1 + (-0.958 - 0.283i)T \)
89 \( 1 + (0.349 + 0.937i)T \)
97 \( 1 + (0.429 - 0.902i)T \)
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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

−18.33784969886118510283291609577, −17.456425317722444848281282061584, −16.7081049003230060278679619457, −16.52839297831680185619657924199, −15.64343731995709517064146624608, −14.828042544445551503619517239933, −14.0009983321229968645540963297, −13.1143640320234380408840381511, −12.99084669513192192846260864317, −11.96016003962456572968219008767, −11.298546108179377951354797769156, −10.87683281494333580485883039041, −10.281672494923532889767255219957, −8.85483012867376466403939089011, −8.419931548509512761083019993340, −7.58328382079621153386431544302, −7.11153130690015404494615013023, −6.417851319660136411527068070794, −5.292692284812016719604370362, −4.92543418076318773635141077455, −3.89512835785669238279954226763, −3.29902199013305337846736911261, −1.934751001157008472211351208314, −1.230489835215030486242299760238, −0.34243621336422134187866376162, 0.75916362479408867762035931195, 2.16558706584778582842693402307, 3.03718777850365571320897891054, 3.731402640756714290487602012781, 4.54838457881247930457765708724, 5.379525686047404587438418737172, 5.71466681897400610225071670358, 6.790549740199823468359517292909, 7.59388372619708020319663837712, 8.271388551362929987839021161048, 9.0094784225557863014437231535, 9.87614107558143649541210252469, 10.60541954568121287424138207538, 11.108722984854844995243421611839, 11.7500788201926268380591449842, 12.50267384383545210702805873317, 12.883843824316055804196070675364, 14.30137202067100716608854149891, 14.87096646050258030787455168624, 15.48879063584400883818141318370, 15.750617002996712720527464213701, 16.60245093881935087694529651928, 17.316694715351928293542176996830, 18.23481724776065316026146249582, 18.51392693988365682822881986697

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