Properties

Degree 1
Conductor 173
Sign $-0.446 + 0.894i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.520 − 0.853i)2-s + (−0.791 − 0.611i)3-s + (−0.457 − 0.889i)4-s + (−0.581 − 0.813i)5-s + (−0.934 + 0.357i)6-s + (−0.976 − 0.217i)7-s + (−0.997 − 0.0729i)8-s + (0.252 + 0.967i)9-s + (−0.997 + 0.0729i)10-s + (−0.0365 + 0.999i)11-s + (−0.181 + 0.983i)12-s + (0.520 − 0.853i)13-s + (−0.694 + 0.719i)14-s + (−0.0365 + 0.999i)15-s + (−0.581 + 0.813i)16-s + (0.744 + 0.667i)17-s + ⋯
L(s,χ)  = 1  + (0.520 − 0.853i)2-s + (−0.791 − 0.611i)3-s + (−0.457 − 0.889i)4-s + (−0.581 − 0.813i)5-s + (−0.934 + 0.357i)6-s + (−0.976 − 0.217i)7-s + (−0.997 − 0.0729i)8-s + (0.252 + 0.967i)9-s + (−0.997 + 0.0729i)10-s + (−0.0365 + 0.999i)11-s + (−0.181 + 0.983i)12-s + (0.520 − 0.853i)13-s + (−0.694 + 0.719i)14-s + (−0.0365 + 0.999i)15-s + (−0.581 + 0.813i)16-s + (0.744 + 0.667i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.446 + 0.894i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.446 + 0.894i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $-0.446 + 0.894i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (29, \cdot )$
Sato-Tate  :  $\mu(43)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 173,\ (0:\ ),\ -0.446 + 0.894i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.2291792694 - 0.3705065984i$
$L(\frac12,\chi)$  $\approx$  $-0.2291792694 - 0.3705065984i$
$L(\chi,1)$  $\approx$  0.3908508347 - 0.5508538086i
$L(1,\chi)$  $\approx$  0.3908508347 - 0.5508538086i

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

−27.7935280327605129834317681979, −27.05065376385986456537990289627, −26.16848494499300986731318923060, −25.46424979494182734983899558416, −23.83579077177882867447383627500, −23.37887632020168697652718567054, −22.397360908683140781055010715582, −21.86648213630734985453089155873, −20.83237497120036925461877135775, −18.96304913864584636931381826572, −18.40571064159574603107297304297, −16.84046265987337990384079664731, −16.280989678591799741573890939628, −15.49944098988134176686557481370, −14.51932579374835233349382303501, −13.35835657025179137956697642563, −12.03640942161822670466390884136, −11.29947036895038711653822552325, −9.91581239633703794857531567767, −8.73103278041064471403212060316, −7.21817772663060994827853880733, −6.30681989838441622334137537336, −5.5011231581134920901449132933, −3.86068240237082673911031828342, −3.32084772738625327314048248962, 0.32579988261481449387275272094, 1.79727544930372175791234891823, 3.529227036679523383831805677783, 4.72987355238364964768630141953, 5.77420430349551996331106872806, 6.99002888941158769720801820497, 8.48464605034275086004210989290, 9.93248990432129351901471678016, 10.838620904616749397397717666742, 12.04652482692755272454484093736, 12.859693184667165943182248523088, 13.109402707770012073711801115152, 14.89621058618613208885535205324, 16.01264223516386176065616664418, 17.09600316316861880867917161780, 18.22049399656791532280430321977, 19.29669029683261192965726827338, 19.935746204497861667740619520, 20.945722660527046805431430358631, 22.279086100246495811353741238104, 23.04280239499570692827156175567, 23.53800017207849807508824707639, 24.582207388968486090092379848538, 25.72931901708642104570610273754, 27.40095304029318970010536309779

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