Properties

Label 1-173-173.85-r0-0-0
Degree $1$
Conductor $173$
Sign $-0.898 - 0.439i$
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  + (−0.581 − 0.813i)2-s + (−0.872 − 0.489i)3-s + (−0.322 + 0.946i)4-s + (−0.791 + 0.611i)5-s + (0.109 + 0.994i)6-s + (0.639 − 0.768i)7-s + (0.957 − 0.288i)8-s + (0.520 + 0.853i)9-s + (0.957 + 0.288i)10-s + (0.989 − 0.145i)11-s + (0.744 − 0.667i)12-s + (−0.581 − 0.813i)13-s + (−0.997 − 0.0729i)14-s + (0.989 − 0.145i)15-s + (−0.791 − 0.611i)16-s + (−0.976 − 0.217i)17-s + ⋯
L(s)  = 1  + (−0.581 − 0.813i)2-s + (−0.872 − 0.489i)3-s + (−0.322 + 0.946i)4-s + (−0.791 + 0.611i)5-s + (0.109 + 0.994i)6-s + (0.639 − 0.768i)7-s + (0.957 − 0.288i)8-s + (0.520 + 0.853i)9-s + (0.957 + 0.288i)10-s + (0.989 − 0.145i)11-s + (0.744 − 0.667i)12-s + (−0.581 − 0.813i)13-s + (−0.997 − 0.0729i)14-s + (0.989 − 0.145i)15-s + (−0.791 − 0.611i)16-s + (−0.976 − 0.217i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.898 - 0.439i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ -0.898 - 0.439i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09149853088 - 0.3951585016i\)
\(L(\frac12)\) \(\approx\) \(0.09149853088 - 0.3951585016i\)
\(L(1)\) \(\approx\) \(0.4196087154 - 0.2943286346i\)
\(L(1)\) \(\approx\) \(0.4196087154 - 0.2943286346i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.581 - 0.813i)T \)
3 \( 1 + (-0.872 - 0.489i)T \)
5 \( 1 + (-0.791 + 0.611i)T \)
7 \( 1 + (0.639 - 0.768i)T \)
11 \( 1 + (0.989 - 0.145i)T \)
13 \( 1 + (-0.581 - 0.813i)T \)
17 \( 1 + (-0.976 - 0.217i)T \)
19 \( 1 + (-0.997 + 0.0729i)T \)
23 \( 1 + (0.989 + 0.145i)T \)
29 \( 1 + (0.109 - 0.994i)T \)
31 \( 1 + (-0.872 + 0.489i)T \)
37 \( 1 + (-0.997 + 0.0729i)T \)
41 \( 1 + (0.639 - 0.768i)T \)
43 \( 1 + (-0.322 - 0.946i)T \)
47 \( 1 + (-0.457 - 0.889i)T \)
53 \( 1 + (0.905 - 0.424i)T \)
59 \( 1 + (-0.181 - 0.983i)T \)
61 \( 1 + (-0.976 + 0.217i)T \)
67 \( 1 + (-0.872 - 0.489i)T \)
71 \( 1 + (0.109 - 0.994i)T \)
73 \( 1 + (0.833 + 0.551i)T \)
79 \( 1 + (-0.457 + 0.889i)T \)
83 \( 1 + (0.252 + 0.967i)T \)
89 \( 1 + (-0.0365 - 0.999i)T \)
97 \( 1 + (0.252 - 0.967i)T \)
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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

−27.6190342723280040240089599810, −27.304846946901805116245638083541, −26.172356703006772630190884439795, −24.71244081664994017072299438714, −24.21577850661069614769756381482, −23.31457613645124306882847745971, −22.30106853995136959221210463434, −21.2881554278631254212542546040, −19.896013325871979133919485127822, −18.98104908232071931155069953114, −17.81367010017930622214483038924, −16.97177942551838871296189353792, −16.29867819082498682848279530058, −15.16280799753969714996322651118, −14.683067664929864660388968480517, −12.73161856837683291696369243888, −11.62222659435367040110623370659, −10.86413181703933776132561658538, −9.20362090953509696326977398539, −8.79874044208386088821553742704, −7.24310667233507557377191735596, −6.231704040664670866210577036236, −4.89706686548221284425358448739, −4.32124343685219489079412284214, −1.50428185134913862133115462051, 0.477427065097297842446775491514, 2.0296205538923591538437925791, 3.71910440048114943832508516600, 4.77720543928152332521282530441, 6.7827643741108944558257065794, 7.47137718180476542449765098155, 8.61965813832657933025151136893, 10.33896980353468294260260383538, 10.995791168919183724969099559457, 11.7306263062549837953597863943, 12.71039921463739532940928507817, 13.89353310199424887956815693153, 15.31378343588537124397714478130, 16.8097602099040267605493882384, 17.37462406716755001687738236903, 18.275724663047346521954507602501, 19.3830135458423157564394789734, 19.88240981762867503023617142565, 21.281479228869463053783754959593, 22.429771802552605864386013629748, 22.90099991077787586823467043572, 24.08600119186744266607284655322, 25.1592656556109574073002595530, 26.64065446422233675499431550029, 27.34965472462228671381725175241

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