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Properties

Label 1-4033-4033.1875-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.359 - 0.933i$
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 + (−0.686 − 0.727i)3-s + (−0.5 + 0.866i)4-s + (0.835 − 0.549i)5-s + (0.286 − 0.957i)6-s + (−0.0581 − 0.998i)7-s − 8-s + (−0.0581 + 0.998i)9-s + (0.893 + 0.448i)10-s + (0.893 − 0.448i)11-s + (0.973 − 0.230i)12-s + (0.835 − 0.549i)13-s + (0.835 − 0.549i)14-s + (−0.973 − 0.230i)15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.686 − 0.727i)3-s + (−0.5 + 0.866i)4-s + (0.835 − 0.549i)5-s + (0.286 − 0.957i)6-s + (−0.0581 − 0.998i)7-s − 8-s + (−0.0581 + 0.998i)9-s + (0.893 + 0.448i)10-s + (0.893 − 0.448i)11-s + (0.973 − 0.230i)12-s + (0.835 − 0.549i)13-s + (0.835 − 0.549i)14-s + (−0.973 − 0.230i)15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.359 - 0.933i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1875, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.359 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.706286711 - 1.171771934i\)
\(L(\frac12)\) \(\approx\) \(1.706286711 - 1.171771934i\)
\(L(1)\) \(\approx\) \(1.285002855 - 0.1114741248i\)
\(L(1)\) \(\approx\) \(1.285002855 - 0.1114741248i\)

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 \( 1 + (-0.686 - 0.727i)T \)
5 \( 1 + (0.835 - 0.549i)T \)
7 \( 1 + (-0.0581 - 0.998i)T \)
11 \( 1 + (0.893 - 0.448i)T \)
13 \( 1 + (0.835 - 0.549i)T \)
17 \( 1 - T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.835 - 0.549i)T \)
31 \( 1 + (-0.597 - 0.802i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (-0.286 - 0.957i)T \)
53 \( 1 + (-0.686 - 0.727i)T \)
59 \( 1 + (-0.973 - 0.230i)T \)
61 \( 1 + (0.993 + 0.116i)T \)
67 \( 1 + (0.973 + 0.230i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.835 - 0.549i)T \)
79 \( 1 + (0.286 + 0.957i)T \)
83 \( 1 + (0.396 + 0.918i)T \)
89 \( 1 + (0.835 - 0.549i)T \)
97 \( 1 + (-0.597 + 0.802i)T \)
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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

−18.559926877575717095787531365636, −17.80672165522841477129159010929, −17.67340867706839730404277042826, −16.49355995962292682011007902793, −15.73344381681919315016865941506, −15.1048929395216200033471851330, −14.44749348150079114153090975741, −13.90795177001707815765259901752, −12.97509757082186992139468555555, −12.32175610811592529495766182960, −11.57370281000702464669239912094, −11.1421039253548744418939959617, −10.49012364535003813422925374572, −9.61716871889536356732893767222, −9.20155816320475763510890456338, −8.74792307522717022517587287279, −6.92468216847301708184087134876, −6.35610257565357409942709338722, −5.810836525983413694690267787911, −5.03715900200988333788406982387, −4.44571976203466440055464778804, −3.39759006934660617902439311390, −2.955424255671424014215513105734, −1.79407226637472515138097072650, −1.261952641157028895887911892070, 0.603393367682748991393879123085, 1.18704127527485947929615774805, 2.387083874439895897637426918096, 3.455716257965218356093169131806, 4.33459949180001943562612661772, 5.04386227496038992875679156391, 5.70789859328277218321316831493, 6.56307767839667818219965514488, 6.69704109361972563502207566312, 7.71142653265202027793935347940, 8.4248667236218316867131227489, 9.11958773847201232744190426325, 9.979402515233180514646262421185, 11.02261765600097387127090397551, 11.49503262549879338374823093063, 12.478811108084299307860010358221, 13.18189983157506617068843891502, 13.46813367655321987572443996162, 13.989367112826816268576605171314, 14.79992648322675284531731482970, 16.024487890072714512155102727994, 16.24694410619978059970761395717, 17.18915854976331043131772414078, 17.35489005182327738840438232383, 17.99684824479572192834564987546

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