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Properties

Label 1-173-173.43-r0-0-0
Degree $1$
Conductor $173$
Sign $-0.543 + 0.839i$
Analytic cond. $0.803408$
Root an. cond. $0.803408$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.543 + 0.839i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ -0.543 + 0.839i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3742148132 + 0.6885010365i\)
\(L(\frac12)\) \(\approx\) \(0.3742148132 + 0.6885010365i\)
\(L(1)\) \(\approx\) \(0.6600652356 + 0.4274691912i\)
\(L(1)\) \(\approx\) \(0.6600652356 + 0.4274691912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.997 + 0.0729i)T \)
3 \( 1 + (0.391 + 0.920i)T \)
5 \( 1 + (0.957 + 0.288i)T \)
7 \( 1 + (-0.791 + 0.611i)T \)
11 \( 1 + (0.109 + 0.994i)T \)
13 \( 1 + (-0.997 + 0.0729i)T \)
17 \( 1 + (-0.581 - 0.813i)T \)
19 \( 1 + (0.744 + 0.667i)T \)
23 \( 1 + (0.109 - 0.994i)T \)
29 \( 1 + (-0.457 + 0.889i)T \)
31 \( 1 + (0.391 - 0.920i)T \)
37 \( 1 + (0.744 + 0.667i)T \)
41 \( 1 + (-0.791 + 0.611i)T \)
43 \( 1 + (0.989 + 0.145i)T \)
47 \( 1 + (-0.0365 + 0.999i)T \)
53 \( 1 + (-0.322 - 0.946i)T \)
59 \( 1 + (0.252 - 0.967i)T \)
61 \( 1 + (-0.581 + 0.813i)T \)
67 \( 1 + (0.391 + 0.920i)T \)
71 \( 1 + (-0.457 + 0.889i)T \)
73 \( 1 + (0.905 + 0.424i)T \)
79 \( 1 + (-0.0365 - 0.999i)T \)
83 \( 1 + (0.833 - 0.551i)T \)
89 \( 1 + (-0.934 - 0.357i)T \)
97 \( 1 + (0.833 + 0.551i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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