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Properties

Label 1-4024-4024.1355-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.999 + 0.0237i$
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  + (−0.951 − 0.307i)3-s + (−0.549 − 0.835i)5-s + (0.977 + 0.211i)7-s + (0.810 + 0.585i)9-s + (−0.772 − 0.635i)11-s + (0.996 + 0.0875i)13-s + (0.265 + 0.964i)15-s + (0.971 + 0.235i)17-s + (−0.289 + 0.957i)19-s + (−0.864 − 0.501i)21-s + (−0.580 − 0.814i)23-s + (−0.395 + 0.918i)25-s + (−0.590 − 0.806i)27-s + (0.838 − 0.544i)29-s + (−0.722 + 0.691i)31-s + ⋯
L(s)  = 1  + (−0.951 − 0.307i)3-s + (−0.549 − 0.835i)5-s + (0.977 + 0.211i)7-s + (0.810 + 0.585i)9-s + (−0.772 − 0.635i)11-s + (0.996 + 0.0875i)13-s + (0.265 + 0.964i)15-s + (0.971 + 0.235i)17-s + (−0.289 + 0.957i)19-s + (−0.864 − 0.501i)21-s + (−0.580 − 0.814i)23-s + (−0.395 + 0.918i)25-s + (−0.590 − 0.806i)27-s + (0.838 − 0.544i)29-s + (−0.722 + 0.691i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.999 + 0.0237i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.999 + 0.0237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.223266683 + 0.01453091528i\)
\(L(\frac12)\) \(\approx\) \(1.223266683 + 0.01453091528i\)
\(L(1)\) \(\approx\) \(0.8424707865 - 0.1269235168i\)
\(L(1)\) \(\approx\) \(0.8424707865 - 0.1269235168i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.951 - 0.307i)T \)
5 \( 1 + (-0.549 - 0.835i)T \)
7 \( 1 + (0.977 + 0.211i)T \)
11 \( 1 + (-0.772 - 0.635i)T \)
13 \( 1 + (0.996 + 0.0875i)T \)
17 \( 1 + (0.971 + 0.235i)T \)
19 \( 1 + (-0.289 + 0.957i)T \)
23 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (0.838 - 0.544i)T \)
31 \( 1 + (-0.722 + 0.691i)T \)
37 \( 1 + (0.747 - 0.663i)T \)
41 \( 1 + (0.993 - 0.112i)T \)
43 \( 1 + (-0.756 + 0.654i)T \)
47 \( 1 + (0.229 + 0.973i)T \)
53 \( 1 + (0.289 + 0.957i)T \)
59 \( 1 + (0.997 + 0.0750i)T \)
61 \( 1 + (-0.539 + 0.842i)T \)
67 \( 1 + (0.265 - 0.964i)T \)
71 \( 1 + (0.337 - 0.941i)T \)
73 \( 1 + (0.496 - 0.868i)T \)
79 \( 1 + (0.905 + 0.424i)T \)
83 \( 1 + (-0.418 + 0.908i)T \)
89 \( 1 + (-0.968 - 0.247i)T \)
97 \( 1 + (0.118 + 0.992i)T \)
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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

−18.26884035375217572360529648373, −17.92647591267211449975280650742, −17.24868306875541894440826825816, −16.35368659195859641739112120565, −15.73017673904579148510603557824, −15.190541836612360621297016607595, −14.588375785783754801715203694182, −13.71597978693883979754514957362, −12.93058660444978014042905108517, −12.08503278515632951918995660357, −11.30089592436025115045780297512, −11.179619892618830011741271870854, −10.24494007724955562520556234091, −9.868357370297772025409403626320, −8.61498058820398895712723943654, −7.84954629102299984471108668014, −7.2398828205007134159646870167, −6.58892894225844086860017606900, −5.61443530279348878732560973469, −5.08966340055597144267389089435, −4.20595232794625724969607082343, −3.63815308093810591684051359533, −2.58630217617303539818320442980, −1.56968861805989354453706702216, −0.553697364498597488119750021380, 0.84519609442465899238668257831, 1.34531906938598187010534041873, 2.348335763019993465581261856180, 3.65901017538539645862257830191, 4.3541385637000241615099094949, 5.065941645467689399446589328408, 5.78839444835366382771308071861, 6.19179132795497055987196861224, 7.50470733367500245180711633546, 8.05827641301089804464754711732, 8.396723619489896571805321837348, 9.44986879697492524593631009741, 10.61066844758151256871818978052, 10.831254695962027942652707683275, 11.6971762115402266741006088007, 12.29297847399941116317111573574, 12.736027504510185031038458630529, 13.62090008326126473290246984253, 14.306324746769815524635875023901, 15.23664905868370410280055963805, 16.06112158592833000955741118173, 16.40463283574597632578217233858, 16.96170201930587894384272099008, 17.99874637091421249203209860125, 18.26910159090534872105440094554

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