Properties

Degree 1
Conductor $ 2^{3} \cdot 3 \cdot 43 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 5-s − 7-s + 11-s − 13-s − 17-s + 19-s − 23-s + 25-s − 29-s + 31-s + 35-s + 37-s − 41-s − 47-s + 49-s + 53-s − 55-s + 59-s + 61-s + 65-s − 67-s + 71-s − 73-s − 77-s + 79-s + 83-s + 85-s + ⋯
L(s,χ)  = 1  − 5-s − 7-s + 11-s − 13-s − 17-s + 19-s − 23-s + 25-s − 29-s + 31-s + 35-s + 37-s − 41-s − 47-s + 49-s + 53-s − 55-s + 59-s + 61-s + 65-s − 67-s + 71-s − 73-s − 77-s + 79-s + 83-s + 85-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1032\)    =    \(2^{3} \cdot 3 \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{1032} (773, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 1032,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8650594054$
$L(\frac12,\chi)$  $\approx$  $0.8650594054$
$L(\chi,1)$  $\approx$  0.7772472199
$L(1,\chi)$  $\approx$  0.7772472199

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

−21.98762405616334812546735880521, −20.49982048151135812219127178543, −19.794790851020972113831467426347, −19.516200456231472633273226203014, −18.60385568680036254346103258629, −17.65989343831352273438006950957, −16.68091806319996478815048801536, −16.15578455060999778359631642180, −15.30410858949227584604459639073, −14.64349764914391919102928528807, −13.616326263235294971337269032484, −12.788259457121924833889646529704, −11.8352918230300261703701709057, −11.56038378211481335327119163205, −10.22593912124172235562461828251, −9.52798112798632142793722844168, −8.70570903024338790985425577944, −7.66061642010509180706178945241, −6.930230103477330682754527896195, −6.18392786483301457122084168954, −4.91565724049318837118752477394, −4.00728602463093897200692681413, −3.30618306361140722779565675680, −2.20262167915976398623733898199, −0.644007791625978482323228385778, 0.644007791625978482323228385778, 2.20262167915976398623733898199, 3.30618306361140722779565675680, 4.00728602463093897200692681413, 4.91565724049318837118752477394, 6.18392786483301457122084168954, 6.930230103477330682754527896195, 7.66061642010509180706178945241, 8.70570903024338790985425577944, 9.52798112798632142793722844168, 10.22593912124172235562461828251, 11.56038378211481335327119163205, 11.8352918230300261703701709057, 12.788259457121924833889646529704, 13.616326263235294971337269032484, 14.64349764914391919102928528807, 15.30410858949227584604459639073, 16.15578455060999778359631642180, 16.68091806319996478815048801536, 17.65989343831352273438006950957, 18.60385568680036254346103258629, 19.516200456231472633273226203014, 19.794790851020972113831467426347, 20.49982048151135812219127178543, 21.98762405616334812546735880521

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