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Properties

Label 1-4033-4033.7-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.998 - 0.0601i$
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  + 2-s + (−0.686 + 0.727i)3-s + 4-s + (−0.835 − 0.549i)5-s + (−0.686 + 0.727i)6-s + (−0.0581 + 0.998i)7-s + 8-s + (−0.0581 − 0.998i)9-s + (−0.835 − 0.549i)10-s + (−0.835 + 0.549i)11-s + (−0.686 + 0.727i)12-s + (−0.835 − 0.549i)13-s + (−0.0581 + 0.998i)14-s + (0.973 − 0.230i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.686 + 0.727i)3-s + 4-s + (−0.835 − 0.549i)5-s + (−0.686 + 0.727i)6-s + (−0.0581 + 0.998i)7-s + 8-s + (−0.0581 − 0.998i)9-s + (−0.835 − 0.549i)10-s + (−0.835 + 0.549i)11-s + (−0.686 + 0.727i)12-s + (−0.835 − 0.549i)13-s + (−0.0581 + 0.998i)14-s + (0.973 − 0.230i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.998 - 0.0601i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.998 - 0.0601i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.616759492 - 0.04868059484i\)
\(L(\frac12)\) \(\approx\) \(1.616759492 - 0.04868059484i\)
\(L(1)\) \(\approx\) \(1.195279745 + 0.1826661810i\)
\(L(1)\) \(\approx\) \(1.195279745 + 0.1826661810i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.686 + 0.727i)T \)
5 \( 1 + (-0.835 - 0.549i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (-0.835 + 0.549i)T \)
13 \( 1 + (-0.835 - 0.549i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.0581 + 0.998i)T \)
31 \( 1 + (0.396 + 0.918i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (-0.686 - 0.727i)T \)
53 \( 1 + (-0.286 - 0.957i)T \)
59 \( 1 + (0.973 - 0.230i)T \)
61 \( 1 + (-0.993 + 0.116i)T \)
67 \( 1 + (-0.686 - 0.727i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.0581 - 0.998i)T \)
79 \( 1 + (0.973 - 0.230i)T \)
83 \( 1 + (0.597 + 0.802i)T \)
89 \( 1 + (0.893 - 0.448i)T \)
97 \( 1 + (0.396 - 0.918i)T \)
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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

−18.74948912051018113628136196151, −17.64338684089121369872665906522, −17.02442923759951763270718824298, −16.32030803335571801505126693145, −15.80676040857312405879273603893, −15.02461261525592280215986059061, −14.13043210253753953097046803161, −13.708577177826576290401181064945, −13.06272953725085353149508963420, −12.22737802812701080537545443442, −11.75170545182847010209678242549, −11.109724046205417570667768424692, −10.50900740150774870035981921545, −9.93384119055827942582166668601, −8.15654097740913405756062747339, −7.700944046108423764201502454240, −7.26247369435839569838034453144, −6.398890174445360495783349871559, −5.91590054707775256741647602860, −4.93887272774913718075773809926, −4.212897189162577900767199296155, −3.60601598575443323517869536568, −2.59749093521266819652408268312, −1.88406117915780365476077287642, −0.71303295383425029410298958588, 0.47642107618618944667480168426, 1.91161684621584469464035382380, 2.92815800049011321600590831883, 3.386222587987768082395487040200, 4.59525582563086409008954998510, 5.0146336352153961948539722819, 5.24770792467869193194383950569, 6.32351941080038710081818641965, 7.10628726134243304759703521008, 7.845785676998143815130495291663, 8.746536021368057208619740320934, 9.63815250193350068657372861347, 10.30404891221353380691871218465, 11.200891709372401787533194745553, 11.70737823580689134548310143645, 12.39404927577438111356004706446, 12.61044078519172889250828025572, 13.6191740726537592029210953248, 14.68912015535494541466825284414, 15.16425375791164944036147935104, 15.78353318484833004387929876499, 16.064302881800540350378059139862, 16.74226968475881356875837194133, 17.90319284944282504751935353172, 18.15258055386240365442511125787

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