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Properties

Label 1-173-173.106-r0-0-0
Degree $1$
Conductor $173$
Sign $0.842 - 0.538i$
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  + (0.833 − 0.551i)2-s + (−0.997 + 0.0729i)3-s + (0.391 − 0.920i)4-s + (−0.694 + 0.719i)5-s + (−0.791 + 0.611i)6-s + (0.520 + 0.853i)7-s + (−0.181 − 0.983i)8-s + (0.989 − 0.145i)9-s + (−0.181 + 0.983i)10-s + (0.639 − 0.768i)11-s + (−0.322 + 0.946i)12-s + (0.833 − 0.551i)13-s + (0.905 + 0.424i)14-s + (0.639 − 0.768i)15-s + (−0.694 − 0.719i)16-s + (0.252 + 0.967i)17-s + ⋯
L(s)  = 1  + (0.833 − 0.551i)2-s + (−0.997 + 0.0729i)3-s + (0.391 − 0.920i)4-s + (−0.694 + 0.719i)5-s + (−0.791 + 0.611i)6-s + (0.520 + 0.853i)7-s + (−0.181 − 0.983i)8-s + (0.989 − 0.145i)9-s + (−0.181 + 0.983i)10-s + (0.639 − 0.768i)11-s + (−0.322 + 0.946i)12-s + (0.833 − 0.551i)13-s + (0.905 + 0.424i)14-s + (0.639 − 0.768i)15-s + (−0.694 − 0.719i)16-s + (0.252 + 0.967i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.842 - 0.538i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.842 - 0.538i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.290898451 - 0.3770714422i\)
\(L(\frac12)\) \(\approx\) \(1.290898451 - 0.3770714422i\)
\(L(1)\) \(\approx\) \(1.220054478 - 0.2761659750i\)
\(L(1)\) \(\approx\) \(1.220054478 - 0.2761659750i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.833 - 0.551i)T \)
3 \( 1 + (-0.997 + 0.0729i)T \)
5 \( 1 + (-0.694 + 0.719i)T \)
7 \( 1 + (0.520 + 0.853i)T \)
11 \( 1 + (0.639 - 0.768i)T \)
13 \( 1 + (0.833 - 0.551i)T \)
17 \( 1 + (0.252 + 0.967i)T \)
19 \( 1 + (0.905 - 0.424i)T \)
23 \( 1 + (0.639 + 0.768i)T \)
29 \( 1 + (-0.791 - 0.611i)T \)
31 \( 1 + (-0.997 - 0.0729i)T \)
37 \( 1 + (0.905 - 0.424i)T \)
41 \( 1 + (0.520 + 0.853i)T \)
43 \( 1 + (0.391 + 0.920i)T \)
47 \( 1 + (0.957 + 0.288i)T \)
53 \( 1 + (-0.872 - 0.489i)T \)
59 \( 1 + (-0.457 + 0.889i)T \)
61 \( 1 + (0.252 - 0.967i)T \)
67 \( 1 + (-0.997 + 0.0729i)T \)
71 \( 1 + (-0.791 - 0.611i)T \)
73 \( 1 + (-0.934 - 0.357i)T \)
79 \( 1 + (0.957 - 0.288i)T \)
83 \( 1 + (-0.0365 + 0.999i)T \)
89 \( 1 + (-0.976 + 0.217i)T \)
97 \( 1 + (-0.0365 - 0.999i)T \)
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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

−27.46198420014577052927936854099, −26.80239013057318386579482541628, −25.2965677475587111152856331313, −24.34688343314773301370424980062, −23.618693813807254156377179909663, −22.999613107688066755791031855081, −22.17201712494019302357958678411, −20.694675456229144288011507286348, −20.37752485698397514248081773983, −18.54591141818011507330688767426, −17.374087414629474262121939521604, −16.55884305235961345392448876155, −16.02256826385816074579988088316, −14.722785544934881752495549695737, −13.58920132175826789004623733782, −12.511723247805909514778450644787, −11.72289918669081503261570828591, −10.922524113988187511877944851572, −9.113098439243086254025134010809, −7.550931357210317137873730838062, −7.00341798520029974101707521285, −5.52752005074402628662155254519, −4.55860631347808384005110235727, −3.83545723138297281430141190911, −1.34426213371105023775636527543, 1.31587272417288866037566359738, 3.13348042571476001086118839115, 4.16989729688696334681720899906, 5.5971459567636482079844913641, 6.19532750585427560268271435285, 7.64200604552984441485032687359, 9.39281071500205702116331346399, 11.027423244547578891579202260531, 11.15720619634701870309356430190, 12.152085939220996906435190662265, 13.23521303592267647251542149472, 14.64950151907042611583513244833, 15.38401052555580728505832448902, 16.28840719593633147525260094743, 17.88492569872350826635180158434, 18.701589430911516280467618840, 19.551656596962633776025229650666, 20.98104108463729075791489703223, 21.88643576808734748421994370337, 22.4129325539941811396976985304, 23.42904519060444875782125056939, 24.09400054165267773291835335024, 25.13678318438020890373171496459, 26.81832051123328188538771131623, 27.79668245459522907574140786125

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