↑

Properties

Label 1-4033-4033.594-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.155 - 0.987i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + i·2-s + (0.0581 + 0.998i)3-s − 4-s + (0.918 − 0.396i)5-s + (−0.998 + 0.0581i)6-s + (−0.993 − 0.116i)7-s − i·8-s + (−0.993 + 0.116i)9-s + (0.396 + 0.918i)10-s + (−0.396 + 0.918i)11-s + (−0.0581 − 0.998i)12-s + (0.918 − 0.396i)13-s + (0.116 − 0.993i)14-s + (0.448 + 0.893i)15-s + 16-s + (0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + i·2-s + (0.0581 + 0.998i)3-s − 4-s + (0.918 − 0.396i)5-s + (−0.998 + 0.0581i)6-s + (−0.993 − 0.116i)7-s − i·8-s + (−0.993 + 0.116i)9-s + (0.396 + 0.918i)10-s + (−0.396 + 0.918i)11-s + (−0.0581 − 0.998i)12-s + (0.918 − 0.396i)13-s + (0.116 − 0.993i)14-s + (0.448 + 0.893i)15-s + 16-s + (0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.155 - 0.987i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (594, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.155 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1455252919 + 0.1244027044i\)
\(L(\frac12)\) \(\approx\) \(-0.1455252919 + 0.1244027044i\)
\(L(1)\) \(\approx\) \(0.6849781424 + 0.6018362108i\)
\(L(1)\) \(\approx\) \(0.6849781424 + 0.6018362108i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (0.0581 + 0.998i)T \)
5 \( 1 + (0.918 - 0.396i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (-0.396 + 0.918i)T \)
13 \( 1 + (0.918 - 0.396i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (0.116 - 0.993i)T \)
31 \( 1 + (-0.727 - 0.686i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (-0.0581 + 0.998i)T \)
53 \( 1 + (-0.835 + 0.549i)T \)
59 \( 1 + (-0.448 - 0.893i)T \)
61 \( 1 + (0.230 + 0.973i)T \)
67 \( 1 + (0.0581 - 0.998i)T \)
71 \( 1 + T \)
73 \( 1 + (0.993 - 0.116i)T \)
79 \( 1 + (-0.448 - 0.893i)T \)
83 \( 1 + (-0.286 + 0.957i)T \)
89 \( 1 + (0.802 + 0.597i)T \)
97 \( 1 + (-0.727 + 0.686i)T \)
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

−18.268911322888272412547337279049, −17.19509313497538770912910732696, −16.761211187534456809726866269461, −15.82918744184084606477579391038, −14.58031339719613684201508573502, −14.07118711167460208341464099047, −13.48519333786672993594772996863, −12.97701405374797412738205913355, −12.482412148537307980029513668473, −11.5709160049200596881120651927, −10.867542896363989988372711339180, −10.37459791330924886476066513448, −9.36458348231578190245640813779, −8.91798953527624611256565112887, −8.225269594211966123093462671048, −7.12241218437079152802242962185, −6.46140417710387750703573748029, −5.6609521392345989936171531266, −5.27082511968059180151010831031, −3.619710337907452529618114582504, −3.237805709795347440076261043223, −2.524851291220453457647539508843, −1.67487291456241680847435114922, −1.00205760613875756055750273142, −0.03342435065743952785805086899, 0.94669021781610765040247425917, 2.31132910616408818584213775496, 3.24751073992907662862732989928, 4.07941107798713214683298720119, 4.70714265621047305874762519137, 5.51627445046091003928205848060, 6.12367955876056342018401629061, 6.589284258355749382245921517549, 7.80703139985274764317874017767, 8.41956945856204889835981886504, 9.31949088934705540074859399238, 9.563527977884796271345756307829, 10.37019732341640798524657138544, 10.79673754927216017503579915278, 12.38275997800994337841193193144, 12.848683254102183606636811602405, 13.4500131358020221283246066570, 14.21250553891031253013789545254, 14.94699088931215837488186717142, 15.46861071688642599787734400896, 16.0992876849554735047070163117, 16.91698651206941837620302912911, 17.01792341101649613472072513864, 17.89244217953023941043800718781, 18.68653760853435698262237542258

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