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Properties

Label 1-173-173.156-r1-0-0
Degree $1$
Conductor $173$
Sign $0.450 - 0.892i$
Analytic cond. $18.5914$
Root an. cond. $18.5914$
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  + (0.889 − 0.457i)2-s + (0.967 − 0.252i)3-s + (0.581 − 0.813i)4-s + (0.946 + 0.322i)5-s + (0.744 − 0.667i)6-s + (0.424 − 0.905i)7-s + (0.145 − 0.989i)8-s + (0.872 − 0.489i)9-s + (0.989 − 0.145i)10-s + (−0.0729 + 0.997i)11-s + (0.357 − 0.934i)12-s + (0.457 + 0.889i)13-s + (−0.0365 − 0.999i)14-s + (0.997 + 0.0729i)15-s + (−0.322 − 0.946i)16-s + (−0.994 + 0.109i)17-s + ⋯
L(s)  = 1  + (0.889 − 0.457i)2-s + (0.967 − 0.252i)3-s + (0.581 − 0.813i)4-s + (0.946 + 0.322i)5-s + (0.744 − 0.667i)6-s + (0.424 − 0.905i)7-s + (0.145 − 0.989i)8-s + (0.872 − 0.489i)9-s + (0.989 − 0.145i)10-s + (−0.0729 + 0.997i)11-s + (0.357 − 0.934i)12-s + (0.457 + 0.889i)13-s + (−0.0365 − 0.999i)14-s + (0.997 + 0.0729i)15-s + (−0.322 − 0.946i)16-s + (−0.994 + 0.109i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.450 - 0.892i$
Analytic conductor: \(18.5914\)
Root analytic conductor: \(18.5914\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (1:\ ),\ 0.450 - 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.469613598 - 2.751667103i\)
\(L(\frac12)\) \(\approx\) \(4.469613598 - 2.751667103i\)
\(L(1)\) \(\approx\) \(2.620138438 - 1.071678739i\)
\(L(1)\) \(\approx\) \(2.620138438 - 1.071678739i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.889 - 0.457i)T \)
3 \( 1 + (0.967 - 0.252i)T \)
5 \( 1 + (0.946 + 0.322i)T \)
7 \( 1 + (0.424 - 0.905i)T \)
11 \( 1 + (-0.0729 + 0.997i)T \)
13 \( 1 + (0.457 + 0.889i)T \)
17 \( 1 + (-0.994 + 0.109i)T \)
19 \( 1 + (-0.999 - 0.0365i)T \)
23 \( 1 + (-0.997 + 0.0729i)T \)
29 \( 1 + (0.744 + 0.667i)T \)
31 \( 1 + (-0.252 + 0.967i)T \)
37 \( 1 + (0.0365 - 0.999i)T \)
41 \( 1 + (-0.905 - 0.424i)T \)
43 \( 1 + (-0.581 - 0.813i)T \)
47 \( 1 + (0.520 + 0.853i)T \)
53 \( 1 + (-0.217 + 0.976i)T \)
59 \( 1 + (0.768 - 0.639i)T \)
61 \( 1 + (0.994 + 0.109i)T \)
67 \( 1 + (-0.252 - 0.967i)T \)
71 \( 1 + (0.667 - 0.744i)T \)
73 \( 1 + (-0.957 + 0.288i)T \)
79 \( 1 + (-0.853 - 0.520i)T \)
83 \( 1 + (-0.791 + 0.611i)T \)
89 \( 1 + (0.694 + 0.719i)T \)
97 \( 1 + (0.611 - 0.791i)T \)
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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

−27.18393595793164612608873466528, −26.05897274790484131048374965434, −25.29474035992531265002867110901, −24.67081378095646335278573221079, −23.93344345726250259611697726072, −22.21405695033582554294534241964, −21.6574905822202098044545786725, −20.90419531373214615384163152065, −20.058651918508622749303943586090, −18.58565803269400511737195212261, −17.50676035441618666908057526173, −16.231728185238648196542420912091, −15.36275084733891422451515906514, −14.52239108752062901814708610664, −13.4579246310443695398433955783, −13.00478473831949471022331560005, −11.50817368263595271942940753257, −10.151429752670921481263098071457, −8.55345518716224750559390780188, −8.31652210606078266637642897444, −6.431311412476142779316405812090, −5.51254500764674242365694503439, −4.3326920780699180432859094874, −2.89397989818142465181746236083, −2.0040568365088884930362936965, 1.60648748963436738071789438646, 2.235569356129031857987029431520, 3.79335964363280798068498314075, 4.704132595809710285437965883153, 6.483614447939021933533641339034, 7.13997537866610187966576045595, 8.867302720716983979231130535238, 10.05595508391318685982557001516, 10.814886293475986049925441559313, 12.37965782063027586879588864949, 13.356533562832556262288337982607, 14.05982750391287446078191404764, 14.67960393557928402734258427334, 15.86548696231927285976502621256, 17.46989547596425348724897302132, 18.45755771476398444763453196906, 19.67618227858693974581558549195, 20.39741775781148026119933751701, 21.21063736523990671210981167094, 21.9939320611566370399310446346, 23.39686511209494110210013993472, 24.01360830272035632612574091429, 25.213901317795447292031703209136, 25.83953932525677733683843722648, 26.91343298151386273324290848454

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