Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.169 + 0.985i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.173 − 0.984i)2-s + (0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (0.802 − 0.597i)5-s + (−0.973 − 0.230i)6-s + (0.396 − 0.918i)7-s + (−0.5 + 0.866i)8-s + (−0.993 − 0.116i)9-s + (−0.448 − 0.893i)10-s + (0.448 − 0.893i)11-s + (−0.396 + 0.918i)12-s + (0.396 − 0.918i)13-s + (−0.835 − 0.549i)14-s + (−0.549 − 0.835i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s,χ)  = 1  + (0.173 − 0.984i)2-s + (0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (0.802 − 0.597i)5-s + (−0.973 − 0.230i)6-s + (0.396 − 0.918i)7-s + (−0.5 + 0.866i)8-s + (−0.993 − 0.116i)9-s + (−0.448 − 0.893i)10-s + (0.448 − 0.893i)11-s + (−0.396 + 0.918i)12-s + (0.396 − 0.918i)13-s + (−0.835 − 0.549i)14-s + (−0.549 − 0.835i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.169 + 0.985i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.169 + 0.985i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.169 + 0.985i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (1050, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.169 + 0.985i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-1.268193747 - 1.068406177i$
$L(\frac12,\chi)$  $\approx$  $-1.268193747 - 1.068406177i$
$L(\chi,1)$  $\approx$  0.3790251180 - 1.129128354i
$L(1,\chi)$  $\approx$  0.3790251180 - 1.129128354i

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

−18.77122288703814806325230716235, −17.994740084398551862024881186062, −17.48276440821028152389940802135, −17.080099380271288077488975133688, −16.03287069557128948246229567815, −15.55582965486615285462805489751, −14.93646124340921333136046456299, −14.517407181041364910822987617541, −13.74008151222186474004140143416, −13.24911755894916922322452821620, −11.995755916988827379238554469821, −11.54467283627682181268762078454, −10.55123145145152253489180694276, −9.72374101018461449485143469325, −9.23036080846978111678662402688, −8.791190575258126852008164089683, −7.86369315440588731530451479662, −6.847059106343823682289891739747, −6.333181083666109286900047291144, −5.53346499241395316412588428068, −4.988152507867483013059217677228, −4.20773916503359905454101550081, −3.46931768793464600702422980180, −2.47001692681201987570319896789, −1.66065657843800032743749952440, 0.430308721429359409514305197356, 1.26829802244784494510471400087, 1.52397633401971400772471607226, 2.83756850318638672502789876063, 3.18294763464017591794035542971, 4.45041164350929620143719701969, 5.01751214000414406580390041391, 5.9995435047680609860132192339, 6.407427233825348649075699813139, 7.635277704754781338085537553574, 8.357324392783378515087786648011, 8.83344356086940378737645905151, 9.699660956397065293880242612347, 10.571793922147839860989315674361, 11.014954622639252922861420908169, 11.94020689523569535027047946652, 12.42048646447583691792415572296, 13.33785395480881614034031528771, 13.582416753354059117183659335571, 14.13044417654798091192175089679, 14.74138272743949577622741293178, 16.207300109233531921172924546123, 16.76384512011352262009432870077, 17.63896686721436423976464408489, 18.02138091438739679552816564769

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