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Properties

Label 1-173-173.29-r0-0-0
Degree $1$
Conductor $173$
Sign $-0.446 + 0.894i$
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  + (0.520 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.446 + 0.894i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ -0.446 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2291792694 - 0.3705065984i\)
\(L(\frac12)\) \(\approx\) \(-0.2291792694 - 0.3705065984i\)
\(L(1)\) \(\approx\) \(0.3908508347 - 0.5508538086i\)
\(L(1)\) \(\approx\) \(0.3908508347 - 0.5508538086i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.520 - 0.853i)T \)
3 \( 1 + (-0.791 - 0.611i)T \)
5 \( 1 + (-0.581 - 0.813i)T \)
7 \( 1 + (-0.976 - 0.217i)T \)
11 \( 1 + (-0.0365 + 0.999i)T \)
13 \( 1 + (0.520 - 0.853i)T \)
17 \( 1 + (0.744 + 0.667i)T \)
19 \( 1 + (-0.694 - 0.719i)T \)
23 \( 1 + (-0.0365 - 0.999i)T \)
29 \( 1 + (-0.934 - 0.357i)T \)
31 \( 1 + (-0.791 + 0.611i)T \)
37 \( 1 + (-0.694 - 0.719i)T \)
41 \( 1 + (-0.976 - 0.217i)T \)
43 \( 1 + (-0.457 + 0.889i)T \)
47 \( 1 + (-0.872 - 0.489i)T \)
53 \( 1 + (0.109 - 0.994i)T \)
59 \( 1 + (0.905 + 0.424i)T \)
61 \( 1 + (0.744 - 0.667i)T \)
67 \( 1 + (-0.791 - 0.611i)T \)
71 \( 1 + (-0.934 - 0.357i)T \)
73 \( 1 + (0.989 - 0.145i)T \)
79 \( 1 + (-0.872 + 0.489i)T \)
83 \( 1 + (-0.322 - 0.946i)T \)
89 \( 1 + (0.391 - 0.920i)T \)
97 \( 1 + (-0.322 + 0.946i)T \)
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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

−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