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Properties

Label 1-4024-4024.397-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.238 - 0.971i$
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  + (−0.713 − 0.700i)3-s + (0.905 − 0.424i)5-s + (0.999 + 0.0250i)7-s + (0.0187 + 0.999i)9-s + (−0.900 − 0.435i)11-s + (0.610 − 0.791i)13-s + (−0.943 − 0.331i)15-s + (−0.977 − 0.211i)17-s + (−0.988 + 0.149i)19-s + (−0.695 − 0.718i)21-s + (0.659 + 0.752i)23-s + (0.640 − 0.768i)25-s + (0.686 − 0.726i)27-s + (−0.384 + 0.923i)29-s + (−0.528 + 0.848i)31-s + ⋯
L(s)  = 1  + (−0.713 − 0.700i)3-s + (0.905 − 0.424i)5-s + (0.999 + 0.0250i)7-s + (0.0187 + 0.999i)9-s + (−0.900 − 0.435i)11-s + (0.610 − 0.791i)13-s + (−0.943 − 0.331i)15-s + (−0.977 − 0.211i)17-s + (−0.988 + 0.149i)19-s + (−0.695 − 0.718i)21-s + (0.659 + 0.752i)23-s + (0.640 − 0.768i)25-s + (0.686 − 0.726i)27-s + (−0.384 + 0.923i)29-s + (−0.528 + 0.848i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.238 - 0.971i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.238 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.328502922 - 1.041837464i\)
\(L(\frac12)\) \(\approx\) \(1.328502922 - 1.041837464i\)
\(L(1)\) \(\approx\) \(1.015753382 - 0.3512801763i\)
\(L(1)\) \(\approx\) \(1.015753382 - 0.3512801763i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.713 - 0.700i)T \)
5 \( 1 + (0.905 - 0.424i)T \)
7 \( 1 + (0.999 + 0.0250i)T \)
11 \( 1 + (-0.900 - 0.435i)T \)
13 \( 1 + (0.610 - 0.791i)T \)
17 \( 1 + (-0.977 - 0.211i)T \)
19 \( 1 + (-0.988 + 0.149i)T \)
23 \( 1 + (0.659 + 0.752i)T \)
29 \( 1 + (-0.384 + 0.923i)T \)
31 \( 1 + (-0.528 + 0.848i)T \)
37 \( 1 + (0.668 + 0.743i)T \)
41 \( 1 + (0.747 + 0.663i)T \)
43 \( 1 + (0.803 - 0.595i)T \)
47 \( 1 + (0.580 - 0.814i)T \)
53 \( 1 + (-0.988 - 0.149i)T \)
59 \( 1 + (0.845 + 0.533i)T \)
61 \( 1 + (-0.337 + 0.941i)T \)
67 \( 1 + (0.943 - 0.331i)T \)
71 \( 1 + (0.974 + 0.223i)T \)
73 \( 1 + (-0.0312 - 0.999i)T \)
79 \( 1 + (0.143 - 0.989i)T \)
83 \( 1 + (0.772 - 0.635i)T \)
89 \( 1 + (-0.301 + 0.953i)T \)
97 \( 1 + (-0.106 - 0.994i)T \)
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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

−18.47044717919791498098862365543, −17.75230795715256853856385593899, −17.33311484033814077274954013733, −16.779717624894108954513086444536, −15.816541114472153676453832745141, −15.2616534832734386952769127754, −14.58115202044805956568767747305, −14.02539111224184974963031538513, −12.989903527973778583949689997233, −12.6228512249335246184910028284, −11.24436734852858551245767974129, −11.07714277132678786808416249213, −10.59876083252342979345130242521, −9.59758062396112529175994969822, −9.13283512323667462265428476142, −8.266120701806475949812132922951, −7.29128714184836354665503883201, −6.44768877487234089807658052368, −5.9395998228802851882272021802, −5.12277027559173660810461455907, −4.44347486562207480858915256184, −3.89991889885136416062914477768, −2.40282126985279682104487772244, −2.13264710126156294517391246936, −0.85716160012443687291256502102, 0.64666380318480530113002613551, 1.484060539394754728721416860836, 2.13492627700585797491626973196, 2.96810196884851177953361347983, 4.34365828483260013292174756676, 5.15108540789934303807368147691, 5.501729777867818569351311934967, 6.2356576391514391322263197684, 7.057647765385442512655730942502, 7.89620975861314132263656129673, 8.50284122839998097149609244790, 9.13791938287884518012835101804, 10.3998462197037444895632651312, 10.80636010874453256250035612372, 11.27742285223877615672323338907, 12.30683600516675060763576685131, 13.00930697704777345920019204678, 13.32701890133259987652787740441, 14.022040770529877925337011185170, 14.909247825975183055083194787493, 15.71689464122731795996992426908, 16.48612234986833899319832637490, 17.12190596097303602327563379053, 17.85402351271906807142126311975, 18.032966818129202027381230899314

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