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Properties

Label 1-4033-4033.2315-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.460 + 0.887i$
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 + (−0.835 − 0.549i)3-s + (0.766 + 0.642i)4-s + (−0.396 + 0.918i)5-s + (−0.597 − 0.802i)6-s + (−0.993 + 0.116i)7-s + (0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.286 − 0.957i)12-s + (−0.597 − 0.802i)13-s + (−0.973 − 0.230i)14-s + (0.835 − 0.549i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (−0.835 − 0.549i)3-s + (0.766 + 0.642i)4-s + (−0.396 + 0.918i)5-s + (−0.597 − 0.802i)6-s + (−0.993 + 0.116i)7-s + (0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (−0.686 + 0.727i)10-s + (−0.686 − 0.727i)11-s + (−0.286 − 0.957i)12-s + (−0.597 − 0.802i)13-s + (−0.973 − 0.230i)14-s + (0.835 − 0.549i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.460 + 0.887i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.460 + 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.300749635 + 0.7904672497i\)
\(L(\frac12)\) \(\approx\) \(1.300749635 + 0.7904672497i\)
\(L(1)\) \(\approx\) \(1.090729943 + 0.2789154964i\)
\(L(1)\) \(\approx\) \(1.090729943 + 0.2789154964i\)

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 \( 1 + (-0.835 - 0.549i)T \)
5 \( 1 + (-0.396 + 0.918i)T \)
7 \( 1 + (-0.993 + 0.116i)T \)
11 \( 1 + (-0.686 - 0.727i)T \)
13 \( 1 + (-0.597 - 0.802i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 - T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.686 - 0.727i)T \)
31 \( 1 + (-0.893 - 0.448i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.396 + 0.918i)T \)
53 \( 1 + (-0.286 - 0.957i)T \)
59 \( 1 + (0.835 - 0.549i)T \)
61 \( 1 + (0.286 + 0.957i)T \)
67 \( 1 + (-0.686 - 0.727i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.286 + 0.957i)T \)
79 \( 1 + (-0.973 + 0.230i)T \)
83 \( 1 + (0.597 + 0.802i)T \)
89 \( 1 + (0.835 + 0.549i)T \)
97 \( 1 + (0.0581 - 0.998i)T \)
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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

−18.54822736977817073036337521371, −17.37612620700302496941954892065, −16.71467570280038273493698718687, −16.17315595848740981495257218449, −15.83283994155262279229708003398, −14.9086535181533940716835676384, −14.440724021245628786121290716076, −13.11754334225340353521580374824, −12.86355105341445835942635817425, −12.165835747442468198972297643550, −11.82285434813138410195916798412, −10.70788670936110101632094368534, −10.33757977022578151153304878569, −9.46701839470614212971195511309, −8.99014230693116998130730216972, −7.49100997155964193462400314389, −6.99348428381084377655290021965, −6.14524576313415677108779330800, −5.40596926399087134826565071172, −4.726997589797839626848477460973, −4.325102684059815995359033647752, −3.468048681373183697153834056787, −2.639729510328751570899252222272, −1.49400188409278648402890643849, −0.51728720145512646280531122609, 0.672196220490067221986995788351, 2.2550856840920765785838618256, 2.79744241997900034799663156699, 3.52476306648535670306934569731, 4.375543806476702986247769638221, 5.502544605768673156383226641943, 5.82679376066219951679093984322, 6.49347046613357073984983308197, 7.30544120431492681835489160811, 7.65771290966839168885754004373, 8.52158608249961265488722132312, 9.97093773460443900344855534019, 10.601536516113782310281153723724, 11.08860174856487325714600842151, 11.881937652115502718547292926063, 12.67975922040551406799439019499, 12.8966698745776251842562109485, 13.71667498944282402372811355283, 14.52283827662464305417016682288, 15.2200394018227615062472206845, 15.865704387265125149034411178051, 16.3730774805790928214604640308, 17.19649984934298525940432051776, 17.691867385263159212410285492624, 18.79237544295921696953001773979

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