Properties

Degree 1
Conductor $ 2^{3} \cdot 47 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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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 + 49-s − 51-s − 53-s + 55-s − 57-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 + 49-s − 51-s − 53-s + 55-s − 57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 376 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 376 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(376\)    =    \(2^{3} \cdot 47\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{376} (93, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 376,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.459062873$
$L(\frac12,\chi)$  $\approx$  $2.459062873$
$L(\chi,1)$  $\approx$  1.296122164
$L(1,\chi)$  $\approx$  1.296122164

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

−24.34558751245487597712159219539, −23.51860699805304730722646395675, −22.52870013639146088332874114546, −21.78272024149242251890653426406, −21.0777720707793354067562806575, −20.23628900720677836640073092565, −18.69082520010943670410857793262, −18.06129140433865603381008124942, −17.37017740543009275860559607016, −16.62064495187312740103043131781, −15.66723875406575564585141686200, −14.27875279205308933604997302389, −13.8261511013688434841316920695, −12.46378967961980692788802999348, −11.715552954314817866357401044599, −10.80714216722378978975589845712, −9.952890420986384996367644213729, −8.93014794515243194058166106566, −7.65061699566650475621003494032, −6.46649995240677302465019574625, −5.70525083839611455638672145240, −4.87044439908837907816198187545, −3.61134668798444228469816939521, −1.719460147707985193868120111502, −1.09416802072735167035160078763, 1.09416802072735167035160078763, 1.719460147707985193868120111502, 3.61134668798444228469816939521, 4.87044439908837907816198187545, 5.70525083839611455638672145240, 6.46649995240677302465019574625, 7.65061699566650475621003494032, 8.93014794515243194058166106566, 9.952890420986384996367644213729, 10.80714216722378978975589845712, 11.715552954314817866357401044599, 12.46378967961980692788802999348, 13.8261511013688434841316920695, 14.27875279205308933604997302389, 15.66723875406575564585141686200, 16.62064495187312740103043131781, 17.37017740543009275860559607016, 18.06129140433865603381008124942, 18.69082520010943670410857793262, 20.23628900720677836640073092565, 21.0777720707793354067562806575, 21.78272024149242251890653426406, 22.52870013639146088332874114546, 23.51860699805304730722646395675, 24.34558751245487597712159219539

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