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Properties

Label 1-4024-4024.229-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.816 + 0.577i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.313698023 + 1.053992358i\)
\(L(\frac12)\) \(\approx\) \(3.313698023 + 1.053992358i\)
\(L(1)\) \(\approx\) \(1.834581444 + 0.2783392133i\)
\(L(1)\) \(\approx\) \(1.834581444 + 0.2783392133i\)

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

−18.16103506123960030667536490778, −18.02774562113982843931223823409, −16.91311652860825874073998398768, −16.41831350489924065824384318842, −15.5701808504911648590562395543, −14.97018317374801538070845155241, −14.26011200676065492113834252459, −13.70414023519499057738068524277, −13.06184697268496489206598784276, −12.304117597002258967176227601513, −11.6863475786655921738638414296, −10.74698610385646178282202906405, −9.925714940898992583990956216312, −9.15676637657083696819162911862, −8.820484538267529954208319338258, −7.938927893684314629880276994282, −7.506356937390315148134971387610, −6.563687040097757367547561514210, −5.50880607810656646713420113977, −4.671255226480260418860331790124, −4.25243217956247725722268545669, −3.45204718617757595143668340902, −2.100819893891661214244355558490, −1.84443863787301003374355246433, −0.92140461393074652450459055751, 1.13376997942340130529986221292, 1.90240058918329136698845271067, 2.61139024687814112972935416777, 3.68210127435963005649781020001, 3.77782698441490508059998559243, 5.046754994986082592925764716817, 6.160640867079161735580037863660, 6.38056373991269455153533409745, 7.71223849378999643559654810482, 8.09370932549904200910408642891, 8.52466897201911218979623317071, 9.580017970324488940830312038401, 10.28767721388302059475854113085, 10.907772413204904783011836648280, 11.48673615649063014275066420463, 12.51772868860037319776841442706, 13.23447467571595846856400850554, 14.082432328164316135557258756865, 14.348933317195616798575316339784, 14.94558489289087901219844165976, 15.627965942240985825599681476627, 16.39749731570324318929147927665, 17.33769911782617487354288350002, 18.07955290090766074369615388684, 18.589064390858878449086267949228

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