Properties

Degree 1
Conductor 751
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 11-s − 12-s + 13-s − 14-s − 15-s + 16-s − 17-s + 18-s + 19-s + 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + ⋯
L(s,χ)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 11-s − 12-s + 13-s − 14-s − 15-s + 16-s − 17-s + 18-s + 19-s + 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(751\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{751} (750, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 751,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.475651110$
$L(\frac12,\chi)$  $\approx$  $3.475651110$
$L(\chi,1)$  $\approx$  1.719575161
$L(1,\chi)$  $\approx$  1.719575161

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

−22.28198916841830033894902371589, −21.72274900066539295365149965366, −20.84349506119742704795718745210, −20.21868953806723586839653617323, −18.82059749824938819310811605770, −18.21821085583733691375949081161, −17.199099861529140717339861911603, −16.33722333858207747833601109482, −15.865707116919839568370487641564, −14.982621439531572525994905343, −13.58563792787864190265431399208, −13.17424204114809419556693821111, −12.702288766198074036947952773173, −11.447364101253125436108840223483, −10.7814023814473117339267544851, −10.06014507030296056953193894296, −9.041192787617262615474630069496, −7.32741203910052758801284487977, −6.70063905430819696672796331255, −5.69633293243583257957336622384, −5.47350152729541018410749015929, −4.23306401456501442707456162858, −3.1292280546105368039070888522, −2.08220168742220260687282574652, −0.83929899312793750932141322795, 0.83929899312793750932141322795, 2.08220168742220260687282574652, 3.1292280546105368039070888522, 4.23306401456501442707456162858, 5.47350152729541018410749015929, 5.69633293243583257957336622384, 6.70063905430819696672796331255, 7.32741203910052758801284487977, 9.041192787617262615474630069496, 10.06014507030296056953193894296, 10.7814023814473117339267544851, 11.447364101253125436108840223483, 12.702288766198074036947952773173, 13.17424204114809419556693821111, 13.58563792787864190265431399208, 14.982621439531572525994905343, 15.865707116919839568370487641564, 16.33722333858207747833601109482, 17.199099861529140717339861911603, 18.21821085583733691375949081161, 18.82059749824938819310811605770, 20.21868953806723586839653617323, 20.84349506119742704795718745210, 21.72274900066539295365149965366, 22.28198916841830033894902371589

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