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Properties

Label 1-4033-4033.76-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.976 + 0.216i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (0.173 − 0.984i)2-s + (−0.173 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (−0.642 − 0.766i)5-s − 6-s + (0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.866 + 0.5i)10-s + (0.866 + 0.5i)11-s + (−0.173 + 0.984i)12-s + (−0.939 − 0.342i)13-s + (−0.5 − 0.866i)14-s + (−0.642 + 0.766i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.173 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (−0.642 − 0.766i)5-s − 6-s + (0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.866 + 0.5i)10-s + (0.866 + 0.5i)11-s + (−0.173 + 0.984i)12-s + (−0.939 − 0.342i)13-s + (−0.5 − 0.866i)14-s + (−0.642 + 0.766i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.976 + 0.216i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.976 + 0.216i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1561578456 - 1.423008981i\)
\(L(\frac12)\) \(\approx\) \(-0.1561578456 - 1.423008981i\)
\(L(1)\) \(\approx\) \(0.5297757925 - 0.8277421529i\)
\(L(1)\) \(\approx\) \(0.5297757925 - 0.8277421529i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (-0.642 - 0.766i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.642 - 0.766i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 - T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.939 - 0.342i)T \)
89 \( 1 + (-0.642 + 0.766i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−18.78474410193385465110718222510, −17.878188321973590522152763205082, −17.34832633894792432673679987598, −16.66525995449385767936517909174, −16.05154991941691149964439304200, −15.32481595744810176707648375568, −14.81874850060203508244276352629, −14.39142116034305969413143160173, −13.917945830530849714191085641272, −12.5237350533994148372654082039, −11.86387383525947555882616125554, −11.4172576792793899706392680919, −10.46050351823162630837970587504, −9.78357154953831602455675830810, −8.87026412471452844668721535323, −8.49755147771773506803996272049, −7.65514426465278258765481086207, −6.789088489267245939923437145706, −6.22290362051543571751089158786, −5.278064390977783202567515377733, −4.73466768377414450488836499196, −4.071680431701411469638751090774, −3.23388915974422681795405948064, −2.57265783629114209813313341509, −0.83863443551745202421278019395, 0.56012754493808398728759245854, 1.30947763096233278625211755858, 1.75507648666581115000240147042, 2.91976370461328205022745222121, 3.73201006064953478318061438756, 4.55606828098509361122131399802, 5.16069909414635098311365492446, 5.86974496326455869617435105754, 7.152633698629242819236777291939, 7.70725613630307597037218909933, 8.23627889709969961517381867611, 9.15008628497338468324974102769, 9.85098797792699302557264195283, 10.713913860561390560965422138630, 11.59727025469000045420926371047, 11.95207377237424968417229969730, 12.393841815511234215123037077522, 13.16668833876028912330903294881, 13.821724957483414507987116730285, 14.62636917937574051895283483657, 14.89588047025041727246323815908, 16.45902875198742302555989220201, 16.90124655514372177765596081265, 17.66149245544730890393209859372, 18.01539936872822029110158079687

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