Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.531 + 0.847i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.642 + 0.766i)2-s + (−0.286 − 0.957i)3-s + (−0.173 − 0.984i)4-s + (−0.0581 + 0.998i)5-s + (0.918 + 0.396i)6-s + (0.893 − 0.448i)7-s + (0.866 + 0.5i)8-s + (−0.835 + 0.549i)9-s + (−0.727 − 0.686i)10-s + (−0.727 + 0.686i)11-s + (−0.893 + 0.448i)12-s + (−0.448 − 0.893i)13-s + (−0.230 + 0.973i)14-s + (0.973 − 0.230i)15-s + (−0.939 + 0.342i)16-s + (−0.342 − 0.939i)17-s + ⋯
L(s,χ)  = 1  + (−0.642 + 0.766i)2-s + (−0.286 − 0.957i)3-s + (−0.173 − 0.984i)4-s + (−0.0581 + 0.998i)5-s + (0.918 + 0.396i)6-s + (0.893 − 0.448i)7-s + (0.866 + 0.5i)8-s + (−0.835 + 0.549i)9-s + (−0.727 − 0.686i)10-s + (−0.727 + 0.686i)11-s + (−0.893 + 0.448i)12-s + (−0.448 − 0.893i)13-s + (−0.230 + 0.973i)14-s + (0.973 − 0.230i)15-s + (−0.939 + 0.342i)16-s + (−0.342 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.531 + 0.847i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.531 + 0.847i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.531 + 0.847i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (787, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (1:\ ),\ 0.531 + 0.847i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.043160380 + 0.5772532710i$
$L(\frac12,\chi)$  $\approx$  $1.043160380 + 0.5772532710i$
$L(\chi,1)$  $\approx$  0.7005204341 + 0.1213802224i
$L(1,\chi)$  $\approx$  0.7005204341 + 0.1213802224i

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

−18.05022413649804366142368134312, −17.36312326436725807069489725194, −17.05126489679000063580792612637, −16.25157598997348765222875206058, −15.69930736668213276318090273477, −15.01300503351486580479855909698, −13.9317505268707929341706704063, −13.36266502254131045019135357452, −12.29838970630197747586040982288, −11.95170935798692379318799254534, −11.1763807349279117537417827237, −10.68933190974193160551239014104, −9.92898298117311003023531423250, −8.97319302742043405234983132299, −8.78118170210665870413437096421, −8.17013736929656272675384807054, −7.19287711203873460670929518694, −5.96570990605826456166211765374, −5.18578960374822463348088553114, −4.42505939935627128103239642329, −4.16066594918126849088062924443, −2.84815276214898239534090306342, −2.28076935397366222582820024563, −1.145816762852858167235737115890, −0.39581676620432752322771242429, 0.58101874486947753265856312313, 1.42129955697447464325303535330, 2.28874755974869613273609590660, 3.01044988410869316869136508125, 4.46508601773564212691285057033, 5.28147063819407484728230038117, 5.75196993225853034178286101805, 6.84779884327348928621255315111, 7.21335702555902373703678687438, 7.829704930164266986099138700322, 8.17287036107536069987212383278, 9.40116873702287374824900499868, 10.21323422048519522395366997311, 10.744719163228328375448579379947, 11.37866675234100114014041649822, 12.12870446721235543710855827293, 13.18686735884643809245416741950, 13.76398080964992655545643419001, 14.46042800272995732248443544873, 14.93327846635645962204538043126, 15.68174280976563931175363858599, 16.493843808528768916382873696555, 17.45454219291330979267548633196, 17.76647021419050730222334641306, 18.13139380671597734670924610970

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