Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $0.894 + 0.447i$
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 + (−0.286 − 0.957i)3-s + (−0.939 − 0.342i)4-s + (0.0581 − 0.998i)5-s + (0.993 − 0.116i)6-s + (0.893 − 0.448i)7-s + (0.5 − 0.866i)8-s + (−0.835 + 0.549i)9-s + (0.973 + 0.230i)10-s + (0.973 − 0.230i)11-s + (−0.0581 + 0.998i)12-s + (−0.893 + 0.448i)13-s + (0.286 + 0.957i)14-s + (−0.973 + 0.230i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s,χ)  = 1  + (−0.173 + 0.984i)2-s + (−0.286 − 0.957i)3-s + (−0.939 − 0.342i)4-s + (0.0581 − 0.998i)5-s + (0.993 − 0.116i)6-s + (0.893 − 0.448i)7-s + (0.5 − 0.866i)8-s + (−0.835 + 0.549i)9-s + (0.973 + 0.230i)10-s + (0.973 − 0.230i)11-s + (−0.0581 + 0.998i)12-s + (−0.893 + 0.448i)13-s + (0.286 + 0.957i)14-s + (−0.973 + 0.230i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.894 + 0.447i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.894 + 0.447i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $0.894 + 0.447i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (3600, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ 0.894 + 0.447i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.113079337 + 0.2631741855i$
$L(\frac12,\chi)$  $\approx$  $1.113079337 + 0.2631741855i$
$L(\chi,1)$  $\approx$  0.8513756112 + 0.01456374892i
$L(1,\chi)$  $\approx$  0.8513756112 + 0.01456374892i

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

−18.274730125576940961915987427773, −17.73499822500573940340676220860, −17.345731484086788481804673799502, −16.658101452517180745749924123760, −15.27289597736013839417066582618, −15.14147554574137674189093410174, −14.29363846524723178246962286235, −13.8835727957057803242707958588, −12.652685529142917008977806711752, −11.897643916412406163977416827822, −11.52711933177957284467252492962, −10.63266336695034168563537223838, −10.48817386678237665936114442291, −9.518285196982822254533696521621, −8.95372706977899740470692854992, −8.260429019752801522438749209393, −7.294107600975248890705252552460, −6.32197597799811752459628532445, −5.49758516587325559564898661808, −4.58640860421964941977000743003, −4.201348590853535689642534351727, −3.28933202448812432495215792, −2.44644130270770285138699494058, −1.94977336716252886493122594483, −0.473359042273017754529531940308, 0.82446559205888330450217623155, 1.40377634936888580900072948515, 2.23395488914089932138998029143, 3.88147828173657193586594934688, 4.53049559648491865502741701467, 5.180561935890786663730119384691, 5.907194991982682837350304622568, 6.66732730862824539517613446717, 7.32454576908248822234248995573, 8.03256529569292759140521974328, 8.50243653323771173391571078815, 9.26852643453299892458159170489, 9.99733220191646695057593991701, 11.07840028544296981062115181981, 11.963785626867290638132018624710, 12.264467278655922354375522897105, 13.27079989352112302097439080936, 14.02861143909153843025932477591, 14.10071260293858927115164049355, 15.07198403721117850749280628556, 16.11411520708135985650960342405, 16.59340005762676473592445755290, 17.29774990925724023247354754206, 17.55605522371812658691060956830, 18.21628712278468454619609585941

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