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Properties

Label 1-4024-4024.325-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.951 - 0.307i$
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  + (−0.560 − 0.828i)3-s + (0.00625 − 0.999i)5-s + (−0.858 − 0.512i)7-s + (−0.372 + 0.928i)9-s + (0.965 + 0.259i)11-s + (−0.713 − 0.700i)13-s + (−0.831 + 0.554i)15-s + (−0.990 + 0.137i)17-s + (0.996 − 0.0875i)19-s + (0.0562 + 0.998i)21-s + (0.852 − 0.523i)23-s + (−0.999 − 0.0125i)25-s + (0.977 − 0.211i)27-s + (0.986 − 0.161i)29-s + (0.192 − 0.981i)31-s + ⋯
L(s)  = 1  + (−0.560 − 0.828i)3-s + (0.00625 − 0.999i)5-s + (−0.858 − 0.512i)7-s + (−0.372 + 0.928i)9-s + (0.965 + 0.259i)11-s + (−0.713 − 0.700i)13-s + (−0.831 + 0.554i)15-s + (−0.990 + 0.137i)17-s + (0.996 − 0.0875i)19-s + (0.0562 + 0.998i)21-s + (0.852 − 0.523i)23-s + (−0.999 − 0.0125i)25-s + (0.977 − 0.211i)27-s + (0.986 − 0.161i)29-s + (0.192 − 0.981i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.951 - 0.307i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.951 - 0.307i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1849449318 - 1.173975794i\)
\(L(\frac12)\) \(\approx\) \(0.1849449318 - 1.173975794i\)
\(L(1)\) \(\approx\) \(0.6944136882 - 0.4957883243i\)
\(L(1)\) \(\approx\) \(0.6944136882 - 0.4957883243i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.560 - 0.828i)T \)
5 \( 1 + (0.00625 - 0.999i)T \)
7 \( 1 + (-0.858 - 0.512i)T \)
11 \( 1 + (0.965 + 0.259i)T \)
13 \( 1 + (-0.713 - 0.700i)T \)
17 \( 1 + (-0.990 + 0.137i)T \)
19 \( 1 + (0.996 - 0.0875i)T \)
23 \( 1 + (0.852 - 0.523i)T \)
29 \( 1 + (0.986 - 0.161i)T \)
31 \( 1 + (0.192 - 0.981i)T \)
37 \( 1 + (-0.731 - 0.682i)T \)
41 \( 1 + (0.994 - 0.0999i)T \)
43 \( 1 + (0.915 + 0.401i)T \)
47 \( 1 + (-0.0312 - 0.999i)T \)
53 \( 1 + (0.996 + 0.0875i)T \)
59 \( 1 + (0.441 + 0.897i)T \)
61 \( 1 + (0.325 - 0.945i)T \)
67 \( 1 + (0.831 + 0.554i)T \)
71 \( 1 + (-0.131 + 0.991i)T \)
73 \( 1 + (-0.993 - 0.112i)T \)
79 \( 1 + (-0.739 - 0.673i)T \)
83 \( 1 + (0.787 - 0.615i)T \)
89 \( 1 + (0.889 - 0.457i)T \)
97 \( 1 + (0.0687 - 0.997i)T \)
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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

−18.88803701130325801517847804716, −17.85834098167916533663926149952, −17.52398431103998227398313990774, −16.6753271335088907285470975208, −15.8904441659732761476438317372, −15.60665718005406695431074319258, −14.7009564196248108083156492083, −14.21697809520566573717901032979, −13.45919232044072622248848018478, −12.27451679836342309311013415331, −11.86817417196780384606208554657, −11.20517355558822046878774141404, −10.53276583006893886101015031678, −9.72343265864823823514622262772, −9.29322944432925154957450408942, −8.67056334541036209306824212466, −7.242152433702745071261792725471, −6.723207504218989342621622452000, −6.2227784324537365101958939370, −5.36416095105595312501037079212, −4.5532782473778788697476684734, −3.67923153651154720107318097084, −3.09393231023311010832189185509, −2.354915383732596885066081397163, −0.98441272861716554117791017472, 0.522774986130069196447930945116, 0.94440459704649959994313745109, 2.062568120879219645824963563210, 2.87769552029567729296139858560, 4.0491478716316903874357798039, 4.669183977960102835571582732325, 5.534910337176094402462562688444, 6.176663058669874863662183615437, 7.047892866766677258723502561533, 7.40951720059754901470105473278, 8.43826706337372072926470995575, 9.099685516954137512486098774233, 9.845467735677470375396467720111, 10.58798760899757926151468847642, 11.53852020420966568145535380328, 12.080260707491601351080643352433, 12.774271700406427563573149076598, 13.15672126369458247603468430523, 13.84252706813485858539432565093, 14.66047316969615917188024575956, 15.75342047643634768039916530227, 16.197918318008518244912681475291, 17.05663775173448698150345942481, 17.34755880340274900441583739208, 17.9220032714652266182975069979

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