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Properties

Label 1-4033-4033.10-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.972 - 0.230i$
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 + (−0.686 + 0.727i)3-s + (−0.173 − 0.984i)4-s + (0.893 − 0.448i)5-s + (0.116 + 0.993i)6-s + (−0.835 − 0.549i)7-s + (−0.866 − 0.5i)8-s + (−0.0581 − 0.998i)9-s + (0.230 − 0.973i)10-s + (0.230 + 0.973i)11-s + (0.835 + 0.549i)12-s + (0.549 − 0.835i)13-s + (−0.957 + 0.286i)14-s + (−0.286 + 0.957i)15-s + (−0.939 + 0.342i)16-s + (0.342 + 0.939i)17-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)2-s + (−0.686 + 0.727i)3-s + (−0.173 − 0.984i)4-s + (0.893 − 0.448i)5-s + (0.116 + 0.993i)6-s + (−0.835 − 0.549i)7-s + (−0.866 − 0.5i)8-s + (−0.0581 − 0.998i)9-s + (0.230 − 0.973i)10-s + (0.230 + 0.973i)11-s + (0.835 + 0.549i)12-s + (0.549 − 0.835i)13-s + (−0.957 + 0.286i)14-s + (−0.286 + 0.957i)15-s + (−0.939 + 0.342i)16-s + (0.342 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.972 - 0.230i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.972 - 0.230i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2786422061 - 2.380180210i\)
\(L(\frac12)\) \(\approx\) \(0.2786422061 - 2.380180210i\)
\(L(1)\) \(\approx\) \(1.058283752 - 0.6746075726i\)
\(L(1)\) \(\approx\) \(1.058283752 - 0.6746075726i\)

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

−18.45822079550934013226654268385, −18.00225390613701829042721960407, −17.05097123915701343213674857805, −16.46634694556244936006577756746, −16.197112630724447657513707417, −15.24304597925440463409029312527, −14.18706192712656218807690508928, −13.86284225818599863347236458361, −13.336334656545397859383091937621, −12.56996616540340675789206113940, −11.89569767523474752930707157101, −11.34783893190411054057843610, −10.42080359097025871387411079528, −9.42681370760969273371184511086, −8.85996668903944534749343712790, −7.89100892887206565944563665383, −7.152504079669119506294466657995, −6.37521803462147068946454081675, −5.97420163383377147470297623064, −5.66159373576996125050992648767, −4.60800462401404406624548747475, −3.54455561416898348060515873675, −2.8178944236901753297006433936, −2.02153645672491911594975328216, −0.91647986634333313520902273890, 0.385426139504014081529958160999, 0.958408013159853079069423322465, 2.01667777003701784911721731948, 2.91558415098678058907144798034, 3.81070202641988386804249892441, 4.39702159500262969912002365442, 5.0528488985943640857071410259, 5.990809681184552608546131088888, 6.223936205750528012395942178818, 7.12968082869581926072785857593, 8.68127257484823631074110325241, 9.20721575181758450500299294527, 10.15492999873398644271802154241, 10.28981113565423843600606273009, 10.831250917856308722462325973180, 12.05865222985709179882745947966, 12.47709488187284586178839671638, 13.00723540937684023254989594456, 13.779674338359385041904993056575, 14.4605582278750683636691940777, 15.38043893820607750802468014971, 15.75685726418510914107212280018, 16.66569533638809691487535484556, 17.49950411040539137574970554610, 17.69541884325799700116959993

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