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Properties

Label 1-4024-4024.123-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.449 + 0.893i$
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  + (−0.999 − 0.0375i)3-s + (−0.180 + 0.983i)5-s + (−0.994 + 0.0999i)7-s + (0.997 + 0.0750i)9-s + (−0.229 − 0.973i)11-s + (0.871 + 0.490i)13-s + (0.217 − 0.976i)15-s + (−0.659 + 0.752i)17-s + (−0.824 − 0.565i)19-s + (0.998 − 0.0625i)21-s + (0.965 − 0.259i)23-s + (−0.934 − 0.355i)25-s + (−0.993 − 0.112i)27-s + (−0.00625 − 0.999i)29-s + (−0.610 − 0.791i)31-s + ⋯
L(s)  = 1  + (−0.999 − 0.0375i)3-s + (−0.180 + 0.983i)5-s + (−0.994 + 0.0999i)7-s + (0.997 + 0.0750i)9-s + (−0.229 − 0.973i)11-s + (0.871 + 0.490i)13-s + (0.217 − 0.976i)15-s + (−0.659 + 0.752i)17-s + (−0.824 − 0.565i)19-s + (0.998 − 0.0625i)21-s + (0.965 − 0.259i)23-s + (−0.934 − 0.355i)25-s + (−0.993 − 0.112i)27-s + (−0.00625 − 0.999i)29-s + (−0.610 − 0.791i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.449 + 0.893i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.449 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2657202983 + 0.4312109398i\)
\(L(\frac12)\) \(\approx\) \(0.2657202983 + 0.4312109398i\)
\(L(1)\) \(\approx\) \(0.6085698492 + 0.1052229919i\)
\(L(1)\) \(\approx\) \(0.6085698492 + 0.1052229919i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.999 - 0.0375i)T \)
5 \( 1 + (-0.180 + 0.983i)T \)
7 \( 1 + (-0.994 + 0.0999i)T \)
11 \( 1 + (-0.229 - 0.973i)T \)
13 \( 1 + (0.871 + 0.490i)T \)
17 \( 1 + (-0.659 + 0.752i)T \)
19 \( 1 + (-0.824 - 0.565i)T \)
23 \( 1 + (0.965 - 0.259i)T \)
29 \( 1 + (-0.00625 - 0.999i)T \)
31 \( 1 + (-0.610 - 0.791i)T \)
37 \( 1 + (-0.977 + 0.211i)T \)
41 \( 1 + (0.971 + 0.235i)T \)
43 \( 1 + (-0.831 + 0.554i)T \)
47 \( 1 + (0.787 + 0.615i)T \)
53 \( 1 + (0.824 - 0.565i)T \)
59 \( 1 + (-0.630 - 0.776i)T \)
61 \( 1 + (-0.192 + 0.981i)T \)
67 \( 1 + (0.217 + 0.976i)T \)
71 \( 1 + (0.620 - 0.784i)T \)
73 \( 1 + (0.992 + 0.124i)T \)
79 \( 1 + (-0.838 + 0.544i)T \)
83 \( 1 + (-0.925 + 0.378i)T \)
89 \( 1 + (-0.337 + 0.941i)T \)
97 \( 1 + (0.910 + 0.412i)T \)
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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

−18.29643355403035840738220463829, −17.36236232180255073179775941992, −16.93254135015280433880386240078, −16.24209063297444995726941611629, −15.610004209687747720770497211, −15.33507333313865719505558025146, −13.976031368339555546462132471158, −13.100674013514865726681089435249, −12.71489589824657439543959534046, −12.303030585465506951779198765750, −11.412567046085703844702307808250, −10.57471194395264859494950469030, −10.153023369239535075497323824740, −9.09593301179642715845095929871, −8.82128710257907382755654447442, −7.51451467603542428758175221597, −7.004547885988812926060107979, −6.22984969545914992048654703821, −5.40538827302632847174407546551, −4.90290099861823584373220121831, −4.05476305515108861196640014280, −3.37623369933037546068466636509, −2.06156039249278763759924442798, −1.19221207263804952626410343328, −0.24412540165232928826524046634, 0.75263696596728873088356762602, 2.030026195525200079900536115319, 2.8793510328263733641053522700, 3.78612430278529317532348244591, 4.28672099359641525634125181276, 5.541497377604107230853413921023, 6.20081115066982515439361066619, 6.5407430491005687509784170716, 7.19724976401924539666351686972, 8.24437006750701147762007922348, 9.03900329934263904681015892523, 9.88227140683459697797773057558, 10.71434350744297359288119896488, 11.03632438657747544179906099185, 11.58548282071466613446518318643, 12.59068262492849847354043068070, 13.231838285641354805377953552566, 13.66142568025402495695209423390, 14.79625613245133608300906975725, 15.52433691129477924136187670767, 15.88022588966185214945293653256, 16.767711724209343044544380340518, 17.19523679608021825972800126085, 18.153352214790847568218014174177, 18.70143368317873393818911569137

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