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Properties

Label 1-4033-4033.3005-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.943 - 0.332i$
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  + (−0.939 − 0.342i)2-s + (−0.597 + 0.802i)3-s + (0.766 + 0.642i)4-s + (0.727 − 0.686i)5-s + (0.835 − 0.549i)6-s + (0.973 − 0.230i)7-s + (−0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (−0.918 + 0.396i)10-s + (0.918 + 0.396i)11-s + (−0.973 + 0.230i)12-s + (0.973 − 0.230i)13-s + (−0.993 − 0.116i)14-s + (0.116 + 0.993i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.597 + 0.802i)3-s + (0.766 + 0.642i)4-s + (0.727 − 0.686i)5-s + (0.835 − 0.549i)6-s + (0.973 − 0.230i)7-s + (−0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (−0.918 + 0.396i)10-s + (0.918 + 0.396i)11-s + (−0.973 + 0.230i)12-s + (0.973 − 0.230i)13-s + (−0.993 − 0.116i)14-s + (0.116 + 0.993i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.943 - 0.332i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.943 - 0.332i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.575404732 - 0.2691708274i\)
\(L(\frac12)\) \(\approx\) \(1.575404732 - 0.2691708274i\)
\(L(1)\) \(\approx\) \(0.9241737626 - 0.06598542296i\)
\(L(1)\) \(\approx\) \(0.9241737626 - 0.06598542296i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.597 + 0.802i)T \)
5 \( 1 + (0.727 - 0.686i)T \)
7 \( 1 + (0.973 - 0.230i)T \)
11 \( 1 + (0.918 + 0.396i)T \)
13 \( 1 + (0.973 - 0.230i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.802 + 0.597i)T \)
31 \( 1 + (0.957 - 0.286i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (0.448 + 0.893i)T \)
53 \( 1 + (-0.957 - 0.286i)T \)
59 \( 1 + (-0.993 + 0.116i)T \)
61 \( 1 + (0.998 - 0.0581i)T \)
67 \( 1 + (-0.727 - 0.686i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.396 + 0.918i)T \)
79 \( 1 + (0.973 + 0.230i)T \)
83 \( 1 + (-0.597 + 0.802i)T \)
89 \( 1 + (-0.998 + 0.0581i)T \)
97 \( 1 + (0.727 - 0.686i)T \)
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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

−18.284980388905766993759658205239, −17.803062028084794061735236782584, −17.27100975954491906862906729788, −16.9508228718057991047485494780, −15.87093471013782467378327880582, −15.18333141916301342389837203084, −14.40210249874919681841532843664, −13.84627752351376071865714215181, −13.18444608011276623344803068503, −12.00628207598521953682511474445, −11.392728236914893231288178184937, −10.972431567939502054945933760959, −10.41729309768555339193709401636, −9.33579507670776682252350311482, −8.68424829216926427483734835703, −8.08199459831241448145669973876, −7.22323865938470202202697580694, −6.49243451783444062246533808972, −6.17869986817236135463016705777, −5.43047479253220495632678518808, −4.5124363107333128640969403787, −3.05131432563620123723891488514, −2.249434479283955150362735129197, −1.40996534329918962309075977452, −1.03303736033308772049723098068, 0.952450993551833683666620391407, 1.18929954930090716166142039738, 2.29683597814868000896728597762, 3.35529627784460892495613552577, 4.250096421825127147644195469928, 4.8464648952322836469902160556, 5.78227187618924765814486778373, 6.43968713223125343453717466479, 7.28253873712015775531156661329, 8.35512495461235457947248116090, 8.820998237504878807307284687465, 9.49101059994282938124652624320, 10.067473589369872171083530515537, 10.886304306983733651510068632834, 11.2610115336844020467451825751, 12.20428847045650436855536394455, 12.49338036253205673914328345728, 13.79817758955388699109681957605, 14.3195534536507200780900758969, 15.34209647333100899955260666641, 15.94376170487172086872036617131, 16.64500073591091604518314796186, 17.13657847645554484643361824929, 17.6924798566831439560647341040, 18.1164957635844298777642076158

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