Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.359 + 0.933i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.359 + 0.933i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (114, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.359 + 0.933i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.706286711 + 1.171771934i$
$L(\frac12,\chi)$  $\approx$  $1.706286711 + 1.171771934i$
$L(\chi,1)$  $\approx$  1.285002855 + 0.1114741248i
$L(1,\chi)$  $\approx$  1.285002855 + 0.1114741248i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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