Properties

Degree 1
Conductor 173
Sign $-0.543 + 0.839i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.997 + 0.0729i)2-s + (0.391 + 0.920i)3-s + (0.989 − 0.145i)4-s + (0.957 + 0.288i)5-s + (−0.457 − 0.889i)6-s + (−0.791 + 0.611i)7-s + (−0.976 + 0.217i)8-s + (−0.694 + 0.719i)9-s + (−0.976 − 0.217i)10-s + (0.109 + 0.994i)11-s + (0.520 + 0.853i)12-s + (−0.997 + 0.0729i)13-s + (0.744 − 0.667i)14-s + (0.109 + 0.994i)15-s + (0.957 − 0.288i)16-s + (−0.581 − 0.813i)17-s + ⋯
L(s,χ)  = 1  + (−0.997 + 0.0729i)2-s + (0.391 + 0.920i)3-s + (0.989 − 0.145i)4-s + (0.957 + 0.288i)5-s + (−0.457 − 0.889i)6-s + (−0.791 + 0.611i)7-s + (−0.976 + 0.217i)8-s + (−0.694 + 0.719i)9-s + (−0.976 − 0.217i)10-s + (0.109 + 0.994i)11-s + (0.520 + 0.853i)12-s + (−0.997 + 0.0729i)13-s + (0.744 − 0.667i)14-s + (0.109 + 0.994i)15-s + (0.957 − 0.288i)16-s + (−0.581 − 0.813i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.543 + 0.839i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.543 + 0.839i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $-0.543 + 0.839i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (43, \cdot )$
Sato-Tate  :  $\mu(43)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 173,\ (0:\ ),\ -0.543 + 0.839i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3742148132 + 0.6885010365i$
$L(\frac12,\chi)$  $\approx$  $0.3742148132 + 0.6885010365i$
$L(\chi,1)$  $\approx$  0.6600652356 + 0.4274691912i
$L(1,\chi)$  $\approx$  0.6600652356 + 0.4274691912i

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

−26.79149719869295048416624888990, −26.25217560671379331817371975390, −25.30197629611157380361296617502, −24.53414425548979262076239765986, −23.81603315758308370586764167133, −22.14781474094132954807678854851, −21.12118694785926407841707693435, −19.80150400951007684476239373596, −19.55143457702760398703586061188, −18.355060198809596629145255873539, −17.3478946496014210667394110328, −16.85777118191767218059376807072, −15.47164855059624913021506920655, −13.996140770326278321004681161211, −13.20133378006837523565227189366, −12.15805529809799750396326503476, −10.833970065623344193700699184833, −9.6204058130999494475478854382, −8.926449927109579076824431959009, −7.65837168177230721999316336894, −6.687308151889457122544097331982, −5.774895824005120753319900887660, −3.28600263631806139750412110159, −2.162158427513114571658314920497, −0.80863215942030924101549360190, 2.18590974562092339132913412303, 2.94570817919533166092251824513, 4.95070321938682231609961473290, 6.19682447873062806271140310891, 7.36255756096654420710997401092, 8.87063555781893413644272089388, 9.76224235813091534165379886815, 9.98651175283379844131314677167, 11.43860885763378351843708240964, 12.74284273628523089910725455541, 14.359370873927630550002429584710, 15.092380541220598438844107347554, 16.15809842722251148002313063048, 16.97664679002080105563098777271, 18.03600234283098289509241018527, 18.97946242209753589653079397654, 20.16783889913197616255000055463, 20.74307745420633348137851145757, 22.04900044145691436225487046778, 22.530007536491037134122588794050, 24.61610967764022034969932444356, 25.24074540606730979370411401266, 25.98913614375544838589585677697, 26.737424332359127830130055106009, 27.70502717207160848253769504845

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