Properties

Degree 1
Conductor $ 2^{2} \cdot 11 $
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 − 13-s − 15-s − 17-s + 19-s − 21-s − 23-s + 25-s − 27-s − 29-s − 31-s + 35-s + 37-s + 39-s − 41-s + 43-s + 45-s − 47-s + 49-s + 51-s + 53-s − 57-s − 59-s − 61-s + ⋯
L(s,χ)  = 1  − 3-s + 5-s + 7-s + 9-s − 13-s − 15-s − 17-s + 19-s − 21-s − 23-s + 25-s − 27-s − 29-s − 31-s + 35-s + 37-s + 39-s − 41-s + 43-s + 45-s − 47-s + 49-s + 51-s + 53-s − 57-s − 59-s − 61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(44\)    =    \(2^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{44} (43, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 44,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7600622838$
$L(\frac12,\chi)$  $\approx$  $0.7600622838$
$L(\chi,1)$  $\approx$  0.9024906449
$L(1,\chi)$  $\approx$  0.9024906449

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

−34.15346729684272106043380578223, −33.60625213311540800317874948376, −32.488723456435195652185644207532, −30.7837343166950765025407660372, −29.60681027223561542267337147912, −28.77893942445047647072544291977, −27.60237248544315232420294111024, −26.442423451163784130338143627684, −24.681483218497409008454757604732, −24.04471027020575246774989117385, −22.32883476081719799605147614760, −21.66563972555785204780870557439, −20.30638125042707154146627647832, −18.269033097875990316507319931584, −17.61972287425101875532373646025, −16.51922470349364623489658320724, −14.83198533873399906847331308472, −13.41643453051203828157973988185, −11.956373727740704105680429338707, −10.74074979750747559997644955886, −9.44566152389131881369715831192, −7.38790752248483336704332902315, −5.81981644686087148920681986787, −4.70920812840911269661655082592, −1.869939273182751683408212296752, 1.869939273182751683408212296752, 4.70920812840911269661655082592, 5.81981644686087148920681986787, 7.38790752248483336704332902315, 9.44566152389131881369715831192, 10.74074979750747559997644955886, 11.956373727740704105680429338707, 13.41643453051203828157973988185, 14.83198533873399906847331308472, 16.51922470349364623489658320724, 17.61972287425101875532373646025, 18.269033097875990316507319931584, 20.30638125042707154146627647832, 21.66563972555785204780870557439, 22.32883476081719799605147614760, 24.04471027020575246774989117385, 24.681483218497409008454757604732, 26.442423451163784130338143627684, 27.60237248544315232420294111024, 28.77893942445047647072544291977, 29.60681027223561542267337147912, 30.7837343166950765025407660372, 32.488723456435195652185644207532, 33.60625213311540800317874948376, 34.15346729684272106043380578223

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