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Properties

Label 1-4033-4033.361-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.0991 + 0.995i$
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  + (0.766 + 0.642i)2-s + (0.173 − 0.984i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (0.766 − 0.642i)6-s + (0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.766 + 0.642i)10-s + (−0.766 − 0.642i)11-s + 12-s + (−0.939 + 0.342i)13-s + (0.766 − 0.642i)14-s + (0.939 + 0.342i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 − 0.984i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (0.766 − 0.642i)6-s + (0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.766 + 0.642i)10-s + (−0.766 − 0.642i)11-s + 12-s + (−0.939 + 0.342i)13-s + (0.766 − 0.642i)14-s + (0.939 + 0.342i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0991 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0991 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0991 + 0.995i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.0991 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.459362189 + 1.321127551i\)
\(L(\frac12)\) \(\approx\) \(1.459362189 + 1.321127551i\)
\(L(1)\) \(\approx\) \(1.355531701 + 0.3525192406i\)
\(L(1)\) \(\approx\) \(1.355531701 + 0.3525192406i\)

Euler product

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

−18.49807210731212512070373238715, −17.38749047702707783073571986719, −17.04603524442055737764604219739, −15.78144346566446890560450155540, −15.42221945093719714210903986006, −15.18926600063940246811773457442, −14.24153254781572243392357982784, −13.46034769646992746034489466831, −12.65516203060632545664778614203, −12.23299305441815304840046465186, −11.57957607490111582428584177829, −10.70413657854445686857795548286, −10.076584884001292770867684191632, −9.44959286523883127232766928475, −8.74805270306771809568219534983, −8.13740082439469669681213104627, −6.98521023090204749831016672588, −5.81265113639397729513468260913, −5.20785820493083629585447951652, −4.83280672835771049555012186066, −4.229740823563581822145516799215, −3.1380277508302648520060093171, −2.56533752513031130230295069767, −1.80276244319798391695263476503, −0.45786274694666536721937995933, 0.910928683655202093021091572602, 2.22800376700741462398963955047, 2.975218818626683540902505354964, 3.34958060361079952061615781324, 4.52004368525953695387299309144, 5.21327341458918912486990146243, 6.22611469333496820822792347540, 6.69208907894530680693050397188, 7.48356654915663177134410935278, 7.705816542885121309016530445255, 8.45518219902394572708202681378, 9.6277787346558545515010917506, 10.53960242264980764229868692817, 11.35482413582725892540431816356, 11.82346135507668487602382711560, 12.63961664324924577774115749339, 13.43166410326460607255676148609, 13.89832236936869474609085481544, 14.40620473665126307141599357357, 14.91700097845929844353376552472, 15.89960420877324888029274765324, 16.585162524932775657071430747295, 17.23609566374046213422601530399, 17.93697211449503557353678328171, 18.506898717512472629364183097224

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