↑

Properties

Label 1-4024-4024.1131-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

Related objects

Downloads

Learn more

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: $\chi_{4024} (1131, \cdot )$
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 \( 1 + (-0.915 - 0.401i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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