Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.685 + 0.728i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.766 − 0.642i)2-s + (−0.993 + 0.116i)3-s + (0.173 + 0.984i)4-s + (0.286 + 0.957i)5-s + (0.835 + 0.549i)6-s + (−0.686 + 0.727i)7-s + (0.5 − 0.866i)8-s + (0.973 − 0.230i)9-s + (0.396 − 0.918i)10-s + (0.396 + 0.918i)11-s + (−0.286 − 0.957i)12-s + (0.686 − 0.727i)13-s + (0.993 − 0.116i)14-s + (−0.396 − 0.918i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s,χ)  = 1  + (−0.766 − 0.642i)2-s + (−0.993 + 0.116i)3-s + (0.173 + 0.984i)4-s + (0.286 + 0.957i)5-s + (0.835 + 0.549i)6-s + (−0.686 + 0.727i)7-s + (0.5 − 0.866i)8-s + (0.973 − 0.230i)9-s + (0.396 − 0.918i)10-s + (0.396 + 0.918i)11-s + (−0.286 − 0.957i)12-s + (0.686 − 0.727i)13-s + (0.993 − 0.116i)14-s + (−0.396 − 0.918i)15-s + (−0.939 + 0.342i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.685 + 0.728i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.685 + 0.728i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.685 + 0.728i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (48, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.685 + 0.728i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7505020484 + 0.3244901272i$
$L(\frac12,\chi)$  $\approx$  $0.7505020484 + 0.3244901272i$
$L(\chi,1)$  $\approx$  0.6011407493 + 0.05181777700i
$L(1,\chi)$  $\approx$  0.6011407493 + 0.05181777700i

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

−18.386217613013054357230499053819, −17.29619158753643292409531001727, −16.953742568966014732397717915757, −16.5132492472041815017486059121, −16.147139752804830013858246633719, −15.32146027324084485300641097297, −14.152102893780041317499007596474, −13.738791078831711007075677665798, −12.831582422492577488086913983928, −12.25828070639068300959367357106, −11.297456011335825207684811996438, −10.64853129591329824572503868301, −10.13177409320100647890727243240, −9.282829488606075180401810504084, −8.67838419299911605772396479608, −7.97856930928945130108276526528, −6.82506088672289304176696120693, −6.6438863881075724706115744600, −5.6769183892554276572674572280, −5.32576612127232896582747171226, −4.25788451991864174619294160595, −3.564321803909688037311216846500, −1.85915287876052901065642654710, −1.17900466659602642673096285315, −0.56824438236847061861235084504, 0.72047232688028167117898061294, 1.74486545613112691174850129423, 2.604821759136470547172529212144, 3.31487672858251184262994474069, 4.11991690869182639519408783992, 5.16365578967348536328421407917, 6.127924372282958750937025239324, 6.55660017047469167300514782092, 7.34338167803388884155724830550, 8.04627657086909480601891882171, 9.361701927194175343649159665984, 9.5729004096008426975582805938, 10.29347375432328055209656231362, 10.9974586686501903756555683734, 11.51282123222517472820701511231, 12.19001232523947898141451567179, 12.96171446917098735492089556679, 13.32214539401300607086328728899, 14.73462927981452947980128677504, 15.4152102672425317804744873614, 15.81815236759353232859780338237, 16.92800182864075117214041493791, 17.24235594548163077767480187948, 17.943878968754577642966504494239, 18.69967309842765080559373397834

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