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Properties

Label 1-4033-4033.13-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.134 + 0.990i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.134 + 0.990i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.134 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5788835426 + 0.6630131476i\)
\(L(\frac12)\) \(\approx\) \(0.5788835426 + 0.6630131476i\)
\(L(1)\) \(\approx\) \(0.7856602422 + 0.08049674249i\)
\(L(1)\) \(\approx\) \(0.7856602422 + 0.08049674249i\)

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

−18.36315295467494700989862588133, −17.54032390178481236358069908915, −16.99943937898396363442577005689, −16.38294796302049596435127978073, −15.558579994976653123713931525906, −15.04500538763704915918983777525, −14.194145937251189388329772329737, −13.3805343432483628263566294706, −12.913794292035472792111113844629, −12.570724346029259347747556515674, −11.4296301973970343371413029835, −10.53374984006847567695593277281, −9.49066139413206043548828032454, −9.15348566772558412798546202053, −8.58882266998365257929742282641, −7.776940031164960259020329271404, −7.11262094116809534621283457249, −6.322421195417094045798975024839, −5.82614036637832401304334751938, −5.13243546846667687538803525236, −4.16590948103917381980778915674, −3.004281509032478723381235915705, −1.95405269935101558931367547347, −1.407898251149546755015285583436, −0.33287769978183966737666518806, 0.92467057006894687903425295360, 2.15221924529325163329265442269, 3.01593982462709418650859176223, 3.504572424781779286676074969539, 3.802666150165062892053939954724, 5.27282974660439478930411514879, 5.703088469342855295035099730, 6.91558034698051405149398493794, 7.627745893878307969258156847194, 8.45128427525592351662046000517, 9.20626245439540547057010809878, 9.85811505426126493297926422441, 10.29733143653555889786094800378, 11.055113723545287121243353782208, 11.228296109324439467896829442832, 12.48691419792081675782177126285, 13.34971578866685384236177267748, 13.76401347491464630993735250627, 14.36598219406458321565634358034, 15.405500800796690257961486421443, 16.08560412242906735510350396417, 16.65447624720001151867552235563, 17.2336389061901074444239600459, 18.22880553658492713396915843802, 18.80201364330311696874178630209

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