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Properties

Label 1-4033-4033.1015-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.731 - 0.681i$
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  + (−0.173 + 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.766 + 0.642i)5-s + (−0.766 + 0.642i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.766 + 0.642i)10-s + (−0.766 − 0.642i)11-s + (−0.5 − 0.866i)12-s + (−0.173 + 0.984i)13-s + (−0.766 + 0.642i)14-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.766 + 0.642i)5-s + (−0.766 + 0.642i)6-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.766 + 0.642i)10-s + (−0.766 − 0.642i)11-s + (−0.5 − 0.866i)12-s + (−0.173 + 0.984i)13-s + (−0.766 + 0.642i)14-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.731 - 0.681i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.731 - 0.681i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6957003183 + 1.767449080i\)
\(L(\frac12)\) \(\approx\) \(-0.6957003183 + 1.767449080i\)
\(L(1)\) \(\approx\) \(0.7112228478 + 1.072728476i\)
\(L(1)\) \(\approx\) \(0.7112228478 + 1.072728476i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (-0.766 - 0.642i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 - T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−18.27824384953956604590708959972, −17.6476336455989217057093073784, −17.03855624470924543623879589486, −16.17967149108838717779273545758, −14.91428543271258000082838226696, −14.49143830105674884664959695755, −13.67068287061441997887230832194, −13.19968474923056900764578688838, −12.68364316369149863138761304553, −12.04902145614160580389363314353, −11.18784152085570703419211993939, −10.17203981391400789918421744246, −9.89394910586294085193160802126, −9.14950848018962622624624330473, −8.13675950061572950313598087435, −7.91584029045738979819450697968, −7.198124783190070489869244712197, −5.76865690811931285471050949815, −5.1940322879614062915709702301, −4.383342604984360027272879349522, −3.45137820119021707797718379935, −2.75392610748975632188668016573, −1.77179193864664288932091412907, −1.48726251564762304903790649355, −0.46335661793657508312398938382, 1.52346136403943506178436277467, 2.2019548479447315795098442735, 3.214196097791242700494799313104, 3.91419607037470951987249943086, 5.04752774103180263959755562934, 5.42734499410875400709767918622, 6.07067253105601890547880604568, 7.23881392854612813972934917012, 7.72468886379864733810025432752, 8.41341124200772831282769858983, 9.2483288587686267759218153214, 9.60739599636098309843974743500, 10.39009853722367160014708863276, 11.0692671952466667782766189875, 12.04763336950288303131305126554, 13.16688535872602356706359507169, 13.8878999243987834511793080533, 14.27213288040056867156174117724, 14.66089470774328579758535604283, 15.50858329460241306726250948034, 16.11663012254293899691693477434, 16.64494557199421683486184835936, 17.52617824093430284463069316481, 18.26761465227220995033629546118, 18.80947358080036211240199090035

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