Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.940 + 0.339i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + (−0.0581 − 0.998i)3-s + 4-s + (−0.396 − 0.918i)5-s + (0.0581 + 0.998i)6-s + (−0.993 − 0.116i)7-s − 8-s + (−0.993 + 0.116i)9-s + (0.396 + 0.918i)10-s + (0.396 − 0.918i)11-s + (−0.0581 − 0.998i)12-s + (−0.396 − 0.918i)13-s + (0.993 + 0.116i)14-s + (−0.893 + 0.448i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (−0.0581 − 0.998i)3-s + 4-s + (−0.396 − 0.918i)5-s + (0.0581 + 0.998i)6-s + (−0.993 − 0.116i)7-s − 8-s + (−0.993 + 0.116i)9-s + (0.396 + 0.918i)10-s + (0.396 − 0.918i)11-s + (−0.0581 − 0.998i)12-s + (−0.396 − 0.918i)13-s + (0.993 + 0.116i)14-s + (−0.893 + 0.448i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.940 + 0.339i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.940 + 0.339i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.940 + 0.339i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (1357, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.940 + 0.339i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.08998329654 + 0.01575404711i$
$L(\frac12,\chi)$  $\approx$  $0.08998329654 + 0.01575404711i$
$L(\chi,1)$  $\approx$  0.3843759719 - 0.2857066697i
$L(1,\chi)$  $\approx$  0.3843759719 - 0.2857066697i

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

−18.414544621984330594243650558770, −17.62647799463635203922475250283, −17.1321743138523828665281460792, −16.36530676887637768294615688463, −15.70339961682189836266951268237, −15.34622524100478965742214881945, −14.60066245253045902556227464871, −14.015084309927651180923157096927, −12.66402423805640126920570991162, −11.90120768479332327781564357065, −11.477106747628113838259578552150, −10.56683415322007711174895633965, −10.04408054311212945894084067288, −9.592383151696515863450829312, −8.94101697759215440396326674971, −8.04584720690936709507035983988, −7.25664911019087411275976486613, −6.37727693057832619921307492310, −6.24181281026381174517276024192, −4.823348359447912397450366282446, −4.00540456094393688476927224944, −3.19414626796118281848321817134, −2.64406854514551678142837666081, −1.661602479043089627538217134886, −0.05139069497151830316991100064, 0.74843390122855521138902675958, 1.292975625333793859738037674773, 2.56861461486167223224444475441, 3.068106555036662723579925453397, 4.07064825290733745096363135921, 5.49869991150387480416102623281, 5.96153859068597990585606308819, 6.663662659769039228833532049547, 7.58855320838236149604091635937, 8.01203430963770649760743034994, 8.662957271088758390292594510657, 9.36697564135307688282549585051, 10.11348883008069122858781538324, 10.89711561122139352216898477658, 11.86023965172853996120030579986, 12.27478773894684370781890649021, 12.69930570374806964195184641668, 13.696065797349486233144717746552, 14.29583162143456687165650206487, 15.43960737351108237706134577159, 16.07772820494273921975081964043, 16.63074606549627506655470117079, 17.09787960530504649732510123690, 17.83446873284710426959537207930, 18.707951453186819483326879442025

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