Properties

Label 1-4033-4033.2792-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.997 - 0.0758i$
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  + (−0.5 + 0.866i)2-s + (0.286 − 0.957i)3-s + (−0.5 − 0.866i)4-s + (−0.448 + 0.893i)5-s + (0.686 + 0.727i)6-s + (−0.835 + 0.549i)7-s + 8-s + (−0.835 − 0.549i)9-s + (−0.549 − 0.835i)10-s + (0.549 − 0.835i)11-s + (−0.973 + 0.230i)12-s + (0.893 + 0.448i)13-s + (−0.0581 − 0.998i)14-s + (0.727 + 0.686i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.286 − 0.957i)3-s + (−0.5 − 0.866i)4-s + (−0.448 + 0.893i)5-s + (0.686 + 0.727i)6-s + (−0.835 + 0.549i)7-s + 8-s + (−0.835 − 0.549i)9-s + (−0.549 − 0.835i)10-s + (0.549 − 0.835i)11-s + (−0.973 + 0.230i)12-s + (0.893 + 0.448i)13-s + (−0.0581 − 0.998i)14-s + (0.727 + 0.686i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.997 - 0.0758i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2792, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.997 - 0.0758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.177241517 - 0.04468916360i\)
\(L(\frac12)\) \(\approx\) \(1.177241517 - 0.04468916360i\)
\(L(1)\) \(\approx\) \(0.8131953785 + 0.1144528170i\)
\(L(1)\) \(\approx\) \(0.8131953785 + 0.1144528170i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.286 - 0.957i)T \)
5 \( 1 + (-0.448 + 0.893i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (0.549 - 0.835i)T \)
13 \( 1 + (0.893 + 0.448i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.998 - 0.0581i)T \)
31 \( 1 + (0.918 + 0.396i)T \)
41 \( 1 + (0.984 - 0.173i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (-0.727 - 0.686i)T \)
53 \( 1 + (0.727 - 0.686i)T \)
59 \( 1 + (-0.686 + 0.727i)T \)
61 \( 1 + (-0.918 + 0.396i)T \)
67 \( 1 + (-0.230 + 0.973i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.0581 - 0.998i)T \)
79 \( 1 + (-0.286 - 0.957i)T \)
83 \( 1 + (-0.396 - 0.918i)T \)
89 \( 1 + (0.549 + 0.835i)T \)
97 \( 1 + (-0.918 + 0.396i)T \)
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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

−18.66805264472287803666052372501, −17.728596139855160414695206552759, −16.93851582768930365456678873146, −16.58812849025386106859036214155, −15.905463369338699135405963286588, −15.35899301733343040116831853791, −14.194326148862559242776542662552, −13.63493664368319021123853727301, −12.838110140250310559109273198832, −12.248357991816791828795931611124, −11.52435147110348446678147823098, −10.809389115016077199165856427542, −10.025406076655168585060486101858, −9.50728411926028734993326380749, −9.11073163937162823362357513119, −8.1067616067337282749128394596, −7.75917068633313948175253750232, −6.6211850315140613839482132999, −5.47690655297211970287730305673, −4.56396223035208908640492121244, −4.16585923843722155842385394036, −3.31323290398749271016889155955, −2.89669419418061989198798103242, −1.48525039758672638871958498904, −0.790348714954434855759978626874, 0.57144023661375156205247812746, 1.412681908160031233366979883021, 2.59948728175925374497770039940, 3.26122853491891397939957355469, 4.086118317059018803155317240698, 5.412774981425573858635231507167, 6.28777792635342804285616495611, 6.49448557377648713873592079764, 7.09131168322985776394702379031, 8.00559681632600308136315719387, 8.72521557638448540968853703318, 8.97917231816610418577029506939, 10.09967181641446992887943980441, 10.84528408160283704173703902547, 11.59454960639473671395198910860, 12.28275545578422818681888209493, 13.48517969113361411637466082639, 13.53826444146266176723354157011, 14.57386867083738438863134778855, 14.98060209676438832084797457966, 15.79455332243757622171032376636, 16.35280102213782622707013291092, 17.2023916283539778658807880020, 17.87857325232662964531452035371, 18.59434034858923388899294470393

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