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Properties

Label 1-4033-4033.1210-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: $\chi_{4033} (1210, \cdot )$
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

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

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