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Properties

Label 1-4033-4033.2261-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.545 - 0.838i$
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 + (−0.835 − 0.549i)3-s + (−0.5 − 0.866i)4-s + (−0.597 − 0.802i)5-s + (−0.893 + 0.448i)6-s + (0.396 − 0.918i)7-s − 8-s + (0.396 + 0.918i)9-s + (−0.993 + 0.116i)10-s + (−0.993 − 0.116i)11-s + (−0.0581 + 0.998i)12-s + (−0.597 − 0.802i)13-s + (−0.597 − 0.802i)14-s + (0.0581 + 0.998i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.835 − 0.549i)3-s + (−0.5 − 0.866i)4-s + (−0.597 − 0.802i)5-s + (−0.893 + 0.448i)6-s + (0.396 − 0.918i)7-s − 8-s + (0.396 + 0.918i)9-s + (−0.993 + 0.116i)10-s + (−0.993 − 0.116i)11-s + (−0.0581 + 0.998i)12-s + (−0.597 − 0.802i)13-s + (−0.597 − 0.802i)14-s + (0.0581 + 0.998i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.545 - 0.838i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.545 - 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02623209310 + 0.01422354717i\)
\(L(\frac12)\) \(\approx\) \(0.02623209310 + 0.01422354717i\)
\(L(1)\) \(\approx\) \(0.3065505045 - 0.5136407098i\)
\(L(1)\) \(\approx\) \(0.3065505045 - 0.5136407098i\)

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 \( 1 + (-0.835 - 0.549i)T \)
5 \( 1 + (-0.597 - 0.802i)T \)
7 \( 1 + (0.396 - 0.918i)T \)
11 \( 1 + (-0.993 - 0.116i)T \)
13 \( 1 + (-0.597 - 0.802i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.597 - 0.802i)T \)
31 \( 1 + (-0.973 + 0.230i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (0.893 + 0.448i)T \)
53 \( 1 + (-0.835 - 0.549i)T \)
59 \( 1 + (0.0581 + 0.998i)T \)
61 \( 1 + (0.686 - 0.727i)T \)
67 \( 1 + (-0.0581 - 0.998i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.597 - 0.802i)T \)
79 \( 1 + (-0.893 - 0.448i)T \)
83 \( 1 + (-0.286 - 0.957i)T \)
89 \( 1 + (-0.597 - 0.802i)T \)
97 \( 1 + (-0.973 - 0.230i)T \)
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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

−18.43737165606149268527240475704, −17.816918415159198827666105356295, −17.156699785986040671774825739236, −16.30126786666405334450015013092, −15.77232149773362769180853994521, −15.24247330187612092624284864207, −14.76730234462430813207919623833, −14.1286243331855038771141037529, −12.936439865129909695731347803151, −12.49765435595160160207691874627, −11.64825471578487538485288663739, −11.20904640145955966892927475973, −10.45139200978949156124104292257, −9.45582611315571210760219091426, −8.75309638703296658263635181797, −8.023235801275978254268766048854, −7.00859366635410812239592084140, −6.73462602500081693804735228809, −5.7778127460144608034010566050, −5.20020835370445786547716200842, −4.40546012595776629702404361923, −3.970744525501182571717692725856, −2.77876328185329277084183303531, −2.18926506607485109908062170018, −0.01553590994866465270011465012, 0.47249020020067632775880247437, 1.60441121065258047352314720813, 2.15453644964391082807962413828, 3.392193712515020943657326131653, 4.28468752556345059173387576632, 4.72109785492435223307879749148, 5.49402833991481914146029596561, 6.06446540766852096569918320661, 7.28743840735844000696365765419, 7.803299976375072398932586258375, 8.52218790577718743614950359243, 9.65614629720881536680178868301, 10.40061279824746939962977053660, 10.95948016612087739846545719814, 11.45073090329986643067240678003, 12.33713737175829829650711716157, 12.78448113036486761730811341403, 13.296247360708078379507747523063, 13.919116699416982802950567038681, 14.95218453425338982097760042229, 15.64287063195188497013036542119, 16.35043014570824682568264598381, 17.23564720974422285089939512344, 17.68170180315796556578081037555, 18.37960031004253003564157437880

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