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Properties

Label 1-4033-4033.1671-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.985 - 0.167i$
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  + (0.342 − 0.939i)2-s + (−0.973 + 0.230i)3-s + (−0.766 − 0.642i)4-s + (−0.549 − 0.835i)5-s + (−0.116 + 0.993i)6-s + (−0.0581 − 0.998i)7-s + (−0.866 + 0.5i)8-s + (0.893 − 0.448i)9-s + (−0.973 + 0.230i)10-s + (0.973 + 0.230i)11-s + (0.893 + 0.448i)12-s + (−0.998 + 0.0581i)13-s + (−0.957 − 0.286i)14-s + (0.727 + 0.686i)15-s + (0.173 + 0.984i)16-s + (0.984 − 0.173i)17-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (−0.973 + 0.230i)3-s + (−0.766 − 0.642i)4-s + (−0.549 − 0.835i)5-s + (−0.116 + 0.993i)6-s + (−0.0581 − 0.998i)7-s + (−0.866 + 0.5i)8-s + (0.893 − 0.448i)9-s + (−0.973 + 0.230i)10-s + (0.973 + 0.230i)11-s + (0.893 + 0.448i)12-s + (−0.998 + 0.0581i)13-s + (−0.957 − 0.286i)14-s + (0.727 + 0.686i)15-s + (0.173 + 0.984i)16-s + (0.984 − 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.985 - 0.167i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.985 - 0.167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1081525177 - 1.283620673i\)
\(L(\frac12)\) \(\approx\) \(0.1081525177 - 1.283620673i\)
\(L(1)\) \(\approx\) \(0.5651971808 - 0.5476009392i\)
\(L(1)\) \(\approx\) \(0.5651971808 - 0.5476009392i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.342 - 0.939i)T \)
3 \( 1 + (-0.973 + 0.230i)T \)
5 \( 1 + (-0.549 - 0.835i)T \)
7 \( 1 + (-0.0581 - 0.998i)T \)
11 \( 1 + (0.973 + 0.230i)T \)
13 \( 1 + (-0.998 + 0.0581i)T \)
17 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (-0.642 - 0.766i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (-0.727 + 0.686i)T \)
31 \( 1 + (0.998 + 0.0581i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (-0.396 + 0.918i)T \)
53 \( 1 + (0.835 - 0.549i)T \)
59 \( 1 + (0.727 + 0.686i)T \)
61 \( 1 + (0.918 - 0.396i)T \)
67 \( 1 + (-0.0581 + 0.998i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.973 - 0.230i)T \)
79 \( 1 + (-0.448 - 0.893i)T \)
83 \( 1 + (-0.286 + 0.957i)T \)
89 \( 1 + (-0.116 + 0.993i)T \)
97 \( 1 + (-0.448 + 0.893i)T \)
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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

−18.66831650896784642707642574288, −17.72429211846837679483024983833, −17.1770805339104657260156034210, −16.64059302200816370111207990203, −15.88453805231152552967827149645, −15.13630057363698310491437272058, −14.786356408693327801142821037548, −14.07483772512305115882649722172, −13.07554346496125961339294442327, −12.364208675317383939721038661436, −11.808862358242827017714984993277, −11.479807706611692028246382121, −10.207120635056351094154397551073, −9.70576966190808553194453099239, −8.68865576253491573415287263387, −7.88115090032092682457194502964, −7.26842207146862911042771705141, −6.61058267097206917588367998675, −5.931706552829892887390466722205, −5.47591580643876010510695327964, −4.53270623570913962477121347934, −3.78893805717253733110068998359, −2.980615879483346759111976983896, −1.88737806108128972541639845476, −0.57316603233148402458707462812, 0.3997520025501705778773849213, 0.92449162934498978986002610947, 1.632592574447347856591330853142, 2.96114576044219170733271328180, 3.91283202454085727700757132662, 4.40688267665233703557552616769, 4.900351507647146995956536139087, 5.64514039665613612205198262886, 6.74144409075206616448816540512, 7.25497895642892240642772643059, 8.4165980578624254118462566169, 9.23336711983973119914726195811, 9.89996821172064399716277025024, 10.42232461057426756561686179293, 11.342838808662289839236021488498, 11.69810925172154921111832556273, 12.48274485298440011987372503894, 12.83678090884035980994631537349, 13.65137353469091474039530654327, 14.68787602592519578045605137909, 14.99844375331341513334633142827, 16.20416283208761847270819076656, 16.738973981866203249681095307914, 17.29471137223181818246499512692, 17.77035890647955470435331336266

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