Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.0463 + 0.998i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + (0.286 + 0.957i)3-s + 4-s + (0.448 + 0.893i)5-s + (−0.286 − 0.957i)6-s + (−0.835 − 0.549i)7-s − 8-s + (−0.835 + 0.549i)9-s + (−0.448 − 0.893i)10-s + (0.448 − 0.893i)11-s + (0.286 + 0.957i)12-s + (−0.893 + 0.448i)13-s + (0.835 + 0.549i)14-s + (−0.727 + 0.686i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (0.286 + 0.957i)3-s + 4-s + (0.448 + 0.893i)5-s + (−0.286 − 0.957i)6-s + (−0.835 − 0.549i)7-s − 8-s + (−0.835 + 0.549i)9-s + (−0.448 − 0.893i)10-s + (0.448 − 0.893i)11-s + (0.286 + 0.957i)12-s + (−0.893 + 0.448i)13-s + (0.835 + 0.549i)14-s + (−0.727 + 0.686i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.0463 + 0.998i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.0463 + 0.998i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.0463 + 0.998i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (2669, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.0463 + 0.998i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5665460261 + 0.5934351091i$
$L(\frac12,\chi)$  $\approx$  $0.5665460261 + 0.5934351091i$
$L(\chi,1)$  $\approx$  0.6196863850 + 0.2544659692i
$L(1,\chi)$  $\approx$  0.6196863850 + 0.2544659692i

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

−18.1087808893120536417205011344, −17.63179094165919300473001986494, −17.201171686058840037171043544196, −16.54125035415617291315675184287, −15.568251606659312522052180461289, −15.098334198784903551890233724095, −14.270119034672590740236345800199, −13.1930095881994130450134017881, −12.668185897393415091240034591437, −12.25554492506381897894187686773, −11.62949726030647229004168404914, −10.46310675925441947584057598931, −9.70728197412383083506582050877, −9.26890892738762561820392602508, −8.538628881087051197351950372577, −8.01559528236196576591103075498, −7.12563518299471754785435434953, −6.4136687034639950508260127749, −6.00372067568885348343696178348, −4.98013084898296055287940799998, −3.82732905482363230890482653876, −2.58909045852899516028996651273, −2.25222907265748501502146919245, −1.439522878465478321973181745315, −0.45919629287857056523763106014, 0.643002446078703396826303341501, 2.28926200077625696617570026119, 2.51566768807339306966734866662, 3.54498338553143221473197988853, 4.07221453532270903501579678162, 5.375513824176802400691719598002, 6.25216468000985731648907486068, 6.68563345893209252003831366087, 7.51677689677652528441429665797, 8.39375706328852671489361953767, 9.1347889145567792855621331037, 9.69162025113969413935935158892, 10.39000827429930501544114733794, 10.57105807458865443737165816228, 11.59279180705767963176443760310, 12.09944088803994744424509046973, 13.45702176589555812814720886018, 14.120260851345967675092288640894, 14.54617300915157601350735957927, 15.52579178820964853608529941794, 15.98063060930822407920719189824, 16.69045921436951563140256552475, 17.185157515706750937160486744973, 17.82614084217897358920152623610, 18.91822659089807699550883109834

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