Properties

Degree 1
Conductor $ 2^{2} \cdot 587 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 3-s − 5-s − 7-s + 9-s + 11-s − 13-s + 15-s + 17-s + 19-s + 21-s + 23-s + 25-s − 27-s + 29-s − 31-s − 33-s + 35-s − 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s − 51-s + 53-s − 55-s + ⋯
L(s,χ)  = 1  − 3-s − 5-s − 7-s + 9-s + 11-s − 13-s + 15-s + 17-s + 19-s + 21-s + 23-s + 25-s − 27-s + 29-s − 31-s − 33-s + 35-s − 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s − 51-s + 53-s − 55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 2348 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 2348 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(2348\)    =    \(2^{2} \cdot 587\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{2348} (2347, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 2348,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7087340314$
$L(\frac12,\chi)$  $\approx$  $0.7087340314$
$L(\chi,1)$  $\approx$  0.6254838274
$L(1,\chi)$  $\approx$  0.6254838274

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

−19.67333724292221522512104597643, −18.823271487742174420767963787949, −18.32105591094818071720742893996, −17.146177800772302876276569876642, −16.743297322115605415930555354776, −16.18642374634967463410625971338, −15.401351577817466511996137763329, −14.762108750652480147983471892516, −13.77548288215325778563294563199, −12.774117679785958328611294948446, −12.08229114941215463140063198330, −11.90492411289250037056081464657, −10.948328795762844351094799278279, −10.07381147378783789382134582730, −9.5357292479733762146963005813, −8.58382957226941129612702433225, −7.33373686458774408774160975251, −7.09173452195590922371763398989, −6.25858657270666129251843534880, −5.25321992522607798208751159344, −4.65757456198944199051348880795, −3.56484144730225331734199743982, −3.13908611128303646846942648589, −1.51198992553563685506780998969, −0.56077036460322639583224562614, 0.56077036460322639583224562614, 1.51198992553563685506780998969, 3.13908611128303646846942648589, 3.56484144730225331734199743982, 4.65757456198944199051348880795, 5.25321992522607798208751159344, 6.25858657270666129251843534880, 7.09173452195590922371763398989, 7.33373686458774408774160975251, 8.58382957226941129612702433225, 9.5357292479733762146963005813, 10.07381147378783789382134582730, 10.948328795762844351094799278279, 11.90492411289250037056081464657, 12.08229114941215463140063198330, 12.774117679785958328611294948446, 13.77548288215325778563294563199, 14.762108750652480147983471892516, 15.401351577817466511996137763329, 16.18642374634967463410625971338, 16.743297322115605415930555354776, 17.146177800772302876276569876642, 18.32105591094818071720742893996, 18.823271487742174420767963787949, 19.67333724292221522512104597643

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