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Properties

Label 1-4033-4033.795-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.957 - 0.287i$
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 − 3-s + (0.173 − 0.984i)4-s + (−0.866 + 0.5i)5-s + (−0.766 + 0.642i)6-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + 9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s + (−0.173 + 0.984i)12-s + 13-s + (−0.939 − 0.342i)14-s + (0.866 − 0.5i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s − 3-s + (0.173 − 0.984i)4-s + (−0.866 + 0.5i)5-s + (−0.766 + 0.642i)6-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + 9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s + (−0.173 + 0.984i)12-s + 13-s + (−0.939 − 0.342i)14-s + (0.866 − 0.5i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.957 - 0.287i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (795, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.957 - 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.115279770 - 0.1640339110i\)
\(L(\frac12)\) \(\approx\) \(1.115279770 - 0.1640339110i\)
\(L(1)\) \(\approx\) \(0.8704990133 - 0.3393578535i\)
\(L(1)\) \(\approx\) \(0.8704990133 - 0.3393578535i\)

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

−18.56963155466271907357401599499, −17.56235507594218102058263405164, −16.81406158903555315969012221137, −16.3663390859242929280216714680, −15.93788674724427623112307476577, −15.09398722408459012445195386191, −14.82851487181349607671014288054, −13.372519444638123137854015389979, −13.014106572690421975377323624040, −12.45961194491925273948839245559, −11.71339353325669933879848791728, −11.21487652178643648498167784292, −10.5955540183966882312114159729, −9.156492288146296260237823449050, −8.611876634756845496637605157230, −8.0666328535255990622518609739, −6.97366341236469689463476801315, −6.30892275842021186389441578480, −5.9114112945063845117695464230, −5.14187299658484802523893541481, −4.276129281898902392800735899991, −3.79171070705807669586400649447, −2.93084540510476838562803665073, −1.67828447242946538915210662471, −0.40620511561526583632854241635, 0.735648416919469647114876219082, 1.53206657620763025287674979709, 2.66943885796504607508507115985, 3.701275574997129979319258039379, 4.08860265728816017827487907487, 4.70219169618799357274599363369, 5.693537222446692540876987557429, 6.44739103405685816611978461740, 7.05618235257508926864554089597, 7.502904579312336674929943545886, 8.95648524577042863981391126315, 9.81699703496156116690123878745, 10.47824594452733471583568307924, 10.97245196609321789189353909100, 11.55853563121855279942081679888, 12.21493397625275810204081528107, 12.81634040500282682470593177767, 13.52465598216390377026668150405, 14.21985442731620645139867968256, 15.13243095288675101059707303212, 15.73658158323735362382957963131, 16.11546424149733978978965351642, 17.15705061175028237794750842225, 17.750795785045098061851903239501, 18.72563711489748633530855701685

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