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Properties

Label 1-4024-4024.1843-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  + (0.910 + 0.412i)3-s + (−0.999 + 0.0375i)5-s + (0.996 − 0.0875i)7-s + (0.659 + 0.752i)9-s + (−0.00625 − 0.999i)11-s + (0.0562 − 0.998i)13-s + (−0.925 − 0.378i)15-s + (−0.677 − 0.735i)17-s + (−0.864 − 0.501i)19-s + (0.943 + 0.331i)21-s + (0.986 + 0.161i)23-s + (0.997 − 0.0750i)25-s + (0.289 + 0.957i)27-s + (0.560 + 0.828i)29-s + (−0.395 + 0.918i)31-s + ⋯
L(s)  = 1  + (0.910 + 0.412i)3-s + (−0.999 + 0.0375i)5-s + (0.996 − 0.0875i)7-s + (0.659 + 0.752i)9-s + (−0.00625 − 0.999i)11-s + (0.0562 − 0.998i)13-s + (−0.925 − 0.378i)15-s + (−0.677 − 0.735i)17-s + (−0.864 − 0.501i)19-s + (0.943 + 0.331i)21-s + (0.986 + 0.161i)23-s + (0.997 − 0.0750i)25-s + (0.289 + 0.957i)27-s + (0.560 + 0.828i)29-s + (−0.395 + 0.918i)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: $\chi_{4024} (1843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.651 - 0.758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.910973252 - 0.8775722095i\)
\(L(\frac12)\) \(\approx\) \(1.910973252 - 0.8775722095i\)
\(L(1)\) \(\approx\) \(1.334872348 - 0.06895316738i\)
\(L(1)\) \(\approx\) \(1.334872348 - 0.06895316738i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.910 + 0.412i)T \)
5 \( 1 + (-0.999 + 0.0375i)T \)
7 \( 1 + (0.996 - 0.0875i)T \)
11 \( 1 + (-0.00625 - 0.999i)T \)
13 \( 1 + (0.0562 - 0.998i)T \)
17 \( 1 + (-0.677 - 0.735i)T \)
19 \( 1 + (-0.864 - 0.501i)T \)
23 \( 1 + (0.986 + 0.161i)T \)
29 \( 1 + (0.560 + 0.828i)T \)
31 \( 1 + (-0.395 + 0.918i)T \)
37 \( 1 + (-0.205 + 0.978i)T \)
41 \( 1 + (-0.824 - 0.565i)T \)
43 \( 1 + (-0.787 - 0.615i)T \)
47 \( 1 + (0.982 + 0.186i)T \)
53 \( 1 + (0.864 - 0.501i)T \)
59 \( 1 + (0.920 - 0.389i)T \)
61 \( 1 + (-0.407 - 0.913i)T \)
67 \( 1 + (-0.925 + 0.378i)T \)
71 \( 1 + (-0.704 + 0.709i)T \)
73 \( 1 + (0.780 - 0.625i)T \)
79 \( 1 + (0.277 - 0.960i)T \)
83 \( 1 + (-0.668 - 0.743i)T \)
89 \( 1 + (0.958 - 0.283i)T \)
97 \( 1 + (-0.915 + 0.401i)T \)
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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

−18.56637797815979192434653432323, −18.13668135945999391982027900517, −17.19350428169172895633045460823, −16.61609615429491950773813443537, −15.47731282348655165691362284393, −14.98006038118382579366161844356, −14.78686123186115631154134680502, −13.84383058368245457660730094304, −13.09122356660977043035878739696, −12.37848715652114800369159966460, −11.82844724778621294154946429557, −11.09607750519140716651026118373, −10.31302301873505377014587539240, −9.3287484897599597615193908303, −8.59197301295660307326642157160, −8.258182791043895231230591029486, −7.33778410466076757770038628380, −7.00070333734897513244898031622, −6.03135291694543939209084659656, −4.65075949451506041415990059825, −4.314743599136164137609368065504, −3.684678685488978598671462613830, −2.43286995160856728746293540355, −1.96860057597573409491739855850, −1.0724090053568990327714126803, 0.559731312708069717800569965881, 1.601853465014740533933346301342, 2.74135978160654122827847366580, 3.238442161222918231298429630765, 4.00609287228191726907141026916, 4.91797033600567767910450729068, 5.2055122310608112482973057692, 6.7293186843500179374968569830, 7.307152465468479583107981023049, 8.09817557843351250818554583954, 8.78282180418020917843383184109, 8.81616356976290441973910237806, 10.3682644990296891069978190568, 10.71713231787476467210369656628, 11.36551897908790873151571430198, 12.147623684345188488162169208544, 13.12971809295783926898109978699, 13.62433097799693726712929756567, 14.45649806984912728395217833590, 15.0318710769745113038867221901, 15.52925404778509499739825751970, 16.126223058202470591330509440679, 16.933914608151513133148823313127, 17.75486572185656157170333051289, 18.5925603519625484926685303562

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