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Properties

Label 1-4033-4033.1412-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.179 + 0.983i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (−0.642 + 0.766i)2-s + (0.835 + 0.549i)3-s + (−0.173 − 0.984i)4-s + (−0.116 − 0.993i)5-s + (−0.957 + 0.286i)6-s + (0.597 + 0.802i)7-s + (0.866 + 0.5i)8-s + (0.396 + 0.918i)9-s + (0.835 + 0.549i)10-s + (−0.835 + 0.549i)11-s + (0.396 − 0.918i)12-s + (0.802 − 0.597i)13-s + (−0.998 − 0.0581i)14-s + (0.448 − 0.893i)15-s + (−0.939 + 0.342i)16-s + (0.342 + 0.939i)17-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (0.835 + 0.549i)3-s + (−0.173 − 0.984i)4-s + (−0.116 − 0.993i)5-s + (−0.957 + 0.286i)6-s + (0.597 + 0.802i)7-s + (0.866 + 0.5i)8-s + (0.396 + 0.918i)9-s + (0.835 + 0.549i)10-s + (−0.835 + 0.549i)11-s + (0.396 − 0.918i)12-s + (0.802 − 0.597i)13-s + (−0.998 − 0.0581i)14-s + (0.448 − 0.893i)15-s + (−0.939 + 0.342i)16-s + (0.342 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.179 + 0.983i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1412, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.179 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.660945845 + 1.385493199i\)
\(L(\frac12)\) \(\approx\) \(1.660945845 + 1.385493199i\)
\(L(1)\) \(\approx\) \(0.9508366281 + 0.4432704205i\)
\(L(1)\) \(\approx\) \(0.9508366281 + 0.4432704205i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.642 + 0.766i)T \)
3 \( 1 + (0.835 + 0.549i)T \)
5 \( 1 + (-0.116 - 0.993i)T \)
7 \( 1 + (0.597 + 0.802i)T \)
11 \( 1 + (-0.835 + 0.549i)T \)
13 \( 1 + (0.802 - 0.597i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
19 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (-0.448 - 0.893i)T \)
31 \( 1 + (-0.802 - 0.597i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (-0.973 + 0.230i)T \)
53 \( 1 + (0.993 - 0.116i)T \)
59 \( 1 + (0.448 - 0.893i)T \)
61 \( 1 + (0.230 - 0.973i)T \)
67 \( 1 + (0.597 - 0.802i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.835 - 0.549i)T \)
79 \( 1 + (0.918 - 0.396i)T \)
83 \( 1 + (-0.0581 + 0.998i)T \)
89 \( 1 + (-0.957 + 0.286i)T \)
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

−18.33617347497483620378832715674, −17.95519076228544004593748822322, −16.89003855088241043404859634835, −16.29803200294042442717454265833, −15.33896466186791693911906451902, −14.5387252825848204240084191876, −13.94596234230305683654263674426, −13.32965667574998798905162011759, −12.83758872500887214216452402640, −11.65287262085749622815849033161, −11.26984101213825547437617685678, −10.5299672399966004670671960889, −10.01747296774961424344871154295, −8.97318983071941605518422226050, −8.45451298177284557943508738013, −7.77431459682369316849657050015, −7.02928223630827293972694764908, −6.763021927395655750903105276142, −5.31425342966927814318717272390, −4.15862414583599089570321646923, −3.49950156301112709274866268172, −2.96373337477420751529245957099, −2.11483992355252114299880911668, −1.39585494475535503886105371034, −0.52112629654426783685880354482, 0.59997095857008004176162702638, 1.82081609309117560845282444061, 2.12956858824853091829442103757, 3.5092118065617591761692590789, 4.41352133557321347716074192105, 5.1186580454318328036593153970, 5.53584289306715456211307638017, 6.52654937641230045067011368202, 7.77950724482441611484342271609, 8.098835214155489482827128567744, 8.564390525524882512704113961061, 9.254166036796270914434265048221, 9.86171525902673453098940051349, 10.73570176320473100916366205729, 11.233064014512716406063770410465, 12.623458747320630068629755069262, 13.02851400204805183578959866176, 13.76859085735731619983417473526, 14.85872460955139012259359635786, 15.1652815879696889333375966934, 15.5416020516365730786533003200, 16.39390794839356166559717758818, 16.979084170997435823464687704566, 17.67728387822258087470298888647, 18.49474916763929949521121383044

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