Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.829 - 0.558i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.939 − 0.342i)2-s + (−0.597 + 0.802i)3-s + (0.766 + 0.642i)4-s + (−0.727 + 0.686i)5-s + (0.835 − 0.549i)6-s + (0.973 − 0.230i)7-s + (−0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.918 − 0.396i)10-s + (−0.918 − 0.396i)11-s + (−0.973 + 0.230i)12-s + (0.973 − 0.230i)13-s + (−0.993 − 0.116i)14-s + (−0.116 − 0.993i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s,χ)  = 1  + (−0.939 − 0.342i)2-s + (−0.597 + 0.802i)3-s + (0.766 + 0.642i)4-s + (−0.727 + 0.686i)5-s + (0.835 − 0.549i)6-s + (0.973 − 0.230i)7-s + (−0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.918 − 0.396i)10-s + (−0.918 − 0.396i)11-s + (−0.973 + 0.230i)12-s + (0.973 − 0.230i)13-s + (−0.993 − 0.116i)14-s + (−0.116 − 0.993i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.829 - 0.558i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.829 - 0.558i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.829 - 0.558i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (1028, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.829 - 0.558i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5755415418 - 0.1758590998i$
$L(\frac12,\chi)$  $\approx$  $0.5755415418 - 0.1758590998i$
$L(\chi,1)$  $\approx$  0.5401461806 + 0.04790775182i
$L(1,\chi)$  $\approx$  0.5401461806 + 0.04790775182i

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

−18.38184815779837190029106133329, −17.95996200124428772716822393814, −17.14305386483515623493158117263, −16.78291047985251149366625727206, −15.877141867104717823514093121733, −15.36840136229211368967758754995, −14.74113181283823709396755269968, −13.620100349612909851442139102826, −12.95066674782816628858909358286, −12.2703118333241674999993360468, −11.39933238142946325062224385666, −11.04757685162778786672479997404, −10.54661289695404043085415677065, −9.1348681268544654156746565744, −8.72713195502670468648027855175, −7.946146265838328502996042278366, −7.53768312630727129538406206512, −6.84675421370505373772016391835, −5.86908549580559985733779074988, −5.21418846053372858267016923285, −4.693122397548142498875641193576, −3.35540459469152684256876022132, −2.06806566561887326322113023358, −1.601322442308915998432310089124, −0.73767564181526933053858296183, 0.37620010088101063218397323698, 1.33743028804088823278503182373, 2.54947121803363997445145431246, 3.36025340722820184684757915295, 3.88681665992600251923586125280, 4.89221629500739916989836625874, 5.692865356846397862891138583121, 6.601335262921446433791902211462, 7.38408433063955058660635385214, 8.065477422734283713892208620096, 8.62187014175487265988491439696, 9.51827443222215418093453237421, 10.418950734037755370695660877866, 10.748570057365571563899802333355, 11.40909901077731540095769997559, 11.66318645098801101551964434803, 12.66956136117853160565905973894, 13.62468712060466153653588596703, 14.664975617055462635438385039272, 15.22977590963028563085485200674, 15.80264084417386500842049141552, 16.513967363946558107972462927056, 16.906155557589410782405530460920, 18.0477205483197290042782353491, 18.36079000408042645439614233329

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