Properties

Label 1-4033-4033.350-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.887 + 0.461i$
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 + (0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.342 − 0.939i)5-s + (−0.939 − 0.342i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s + (0.5 + 0.866i)12-s + (−0.766 − 0.642i)13-s + (0.939 + 0.342i)14-s + (−0.642 − 0.766i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.342 − 0.939i)5-s + (−0.939 − 0.342i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s + (0.5 + 0.866i)12-s + (−0.766 − 0.642i)13-s + (0.939 + 0.342i)14-s + (−0.642 − 0.766i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.887 + 0.461i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (350, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.887 + 0.461i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1808164543 - 0.7401300877i\)
\(L(\frac12)\) \(\approx\) \(-0.1808164543 - 0.7401300877i\)
\(L(1)\) \(\approx\) \(0.6395703065 - 0.4672474606i\)
\(L(1)\) \(\approx\) \(0.6395703065 - 0.4672474606i\)

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 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (0.342 + 0.939i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.984 + 0.173i)T \)
31 \( 1 + (0.984 + 0.173i)T \)
41 \( 1 + (0.642 + 0.766i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.642 - 0.766i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (-0.342 - 0.939i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.642 + 0.766i)T \)
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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

−18.84142235970849354809415105519, −18.673517646827844159468390741679, −17.136470926372490244738919556560, −16.92466372817690617019182231048, −16.05669502042867685800410746482, −15.508733867989889041955515299951, −14.799628346542549909833363600535, −14.335025375034966142204803432135, −13.7653476078493428195402182500, −12.92710620061187166735827025564, −11.84263111101649239172186550901, −10.92846767702671642502288133426, −10.32659670706537410542848160085, −9.87678137353065910735202332844, −9.03545292546157747248382962979, −8.53899008667501484352745418152, −7.48691704937747104424570983128, −7.31958065936430578237590473140, −6.3073419086393351666366006053, −5.85506242057366303060119567272, −4.48052353266301347481227607682, −3.778630532198035603313098466524, −3.03866926753796962460184505311, −2.249982640825193384611974271076, −1.229558247199084794278647562558, 0.26325560681308780525241315129, 1.13767994590661152962810324162, 2.089729843528417096481714990802, 2.85005141646583390685996097197, 3.35732277459670030663747788949, 4.41108021223099095982378743779, 4.99387392829601579656099790753, 6.440310648254974727268134305456, 7.309225995259648168633451173810, 7.53709629417755724156686081836, 8.64226345457217498189385988579, 9.02859318001415579980447401561, 9.70022682329902931078546047915, 10.02018917376978630219562036127, 11.34577241647896328716997280605, 12.04035700256911967549122359566, 12.63399539662244414869830412773, 13.02999088283004526861720841440, 13.628766887963634905947494604127, 14.92241744070447669617700432364, 15.38991245697719539939974339566, 16.07668033554885520626504112632, 16.86072308234080515632302954613, 17.4802518310173117493218777818, 18.23207073165279733772362186902

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