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Properties

Label 1-4033-4033.3-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.743 - 0.668i$
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  + (0.5 + 0.866i)2-s + (0.396 − 0.918i)3-s + (−0.5 + 0.866i)4-s + (0.286 + 0.957i)5-s + (0.993 − 0.116i)6-s + (−0.686 + 0.727i)7-s − 8-s + (−0.686 − 0.727i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (0.597 + 0.802i)12-s + (0.286 + 0.957i)13-s + (−0.973 − 0.230i)14-s + (0.993 + 0.116i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.396 − 0.918i)3-s + (−0.5 + 0.866i)4-s + (0.286 + 0.957i)5-s + (0.993 − 0.116i)6-s + (−0.686 + 0.727i)7-s − 8-s + (−0.686 − 0.727i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (0.597 + 0.802i)12-s + (0.286 + 0.957i)13-s + (−0.973 − 0.230i)14-s + (0.993 + 0.116i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.743 - 0.668i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.743 - 0.668i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8243863964 - 0.3163566959i\)
\(L(\frac12)\) \(\approx\) \(0.8243863964 - 0.3163566959i\)
\(L(1)\) \(\approx\) \(1.018012077 + 0.3926207350i\)
\(L(1)\) \(\approx\) \(1.018012077 + 0.3926207350i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.396 - 0.918i)T \)
5 \( 1 + (0.286 + 0.957i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (-0.686 - 0.727i)T \)
13 \( 1 + (0.286 + 0.957i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.973 - 0.230i)T \)
31 \( 1 + (0.0581 + 0.998i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.993 - 0.116i)T \)
53 \( 1 + (-0.993 + 0.116i)T \)
59 \( 1 + (0.993 + 0.116i)T \)
61 \( 1 + (0.0581 - 0.998i)T \)
67 \( 1 + (0.597 - 0.802i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.973 - 0.230i)T \)
79 \( 1 + (-0.396 - 0.918i)T \)
83 \( 1 + (-0.0581 + 0.998i)T \)
89 \( 1 + (0.686 - 0.727i)T \)
97 \( 1 + (0.0581 - 0.998i)T \)
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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

−18.91504925186590722315332210574, −17.56718534040793987231578423917, −17.40190659835688454427359029069, −16.35850369341913305266428742955, −15.70832244735838873971883695546, −15.1278486888189619365604185524, −14.43449075080146708696289984039, −13.5124129956006334946011634720, −12.896775623990602557696790146020, −12.84845315846259466352781318028, −11.62373589095840295526598846588, −10.72254826797118075072679856665, −10.3627846790235947381146219807, −9.57905933945958405218905495971, −9.27842504262538561990817389208, −8.247794657238698712441256484, −7.61658880828122269764207167086, −6.187777343399747971695793044176, −5.52686285341374536879262846306, −4.94745345829705974941881204639, −4.037483753736483894777923281170, −3.748638618110832174864625047191, −2.73412236273801211870935536155, −2.00395027898716687798748221612, −0.93384704084558841857074694930, 0.21016859418282184051394534650, 1.89680926105503273660924595844, 2.69275053031149976153577237322, 3.17210517487001850268966488425, 3.9641555306669585948130858171, 5.250381449863415040929517624, 5.99183523559947291184134855371, 6.4167921788327844843831995451, 6.987460459140059220274425361272, 7.79853087155955381927582257903, 8.440952660663560355171024866620, 9.16005260270688104254190314073, 9.84062470078213728272513089844, 11.04743031673124404644439605421, 11.70647203102072291922507951281, 12.570778844677085520004669969768, 12.964386799843844147570320938105, 13.856200608588055290182789556840, 14.23383265775796575810466969552, 14.774446984236158118458212702331, 15.63103135419496548974402023384, 16.28059395653016847269610984534, 16.89409954592076332218260573378, 17.95400942685415440570390078420, 18.358989635202519326138085986906

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