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Properties

Label 1-173-173.9-r0-0-0
Degree $1$
Conductor $173$
Sign $-0.249 + 0.968i$
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  + (−0.391 + 0.920i)2-s + (−0.989 + 0.145i)3-s + (−0.694 − 0.719i)4-s + (0.0365 + 0.999i)5-s + (0.252 − 0.967i)6-s + (0.457 − 0.889i)7-s + (0.934 − 0.357i)8-s + (0.957 − 0.288i)9-s + (−0.934 − 0.357i)10-s + (0.181 + 0.983i)11-s + (0.791 + 0.611i)12-s + (0.391 − 0.920i)13-s + (0.639 + 0.768i)14-s + (−0.181 − 0.983i)15-s + (−0.0365 + 0.999i)16-s + (0.872 − 0.489i)17-s + ⋯
L(s)  = 1  + (−0.391 + 0.920i)2-s + (−0.989 + 0.145i)3-s + (−0.694 − 0.719i)4-s + (0.0365 + 0.999i)5-s + (0.252 − 0.967i)6-s + (0.457 − 0.889i)7-s + (0.934 − 0.357i)8-s + (0.957 − 0.288i)9-s + (−0.934 − 0.357i)10-s + (0.181 + 0.983i)11-s + (0.791 + 0.611i)12-s + (0.391 − 0.920i)13-s + (0.639 + 0.768i)14-s + (−0.181 − 0.983i)15-s + (−0.0365 + 0.999i)16-s + (0.872 − 0.489i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.249 + 0.968i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ -0.249 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4128567149 + 0.5325130346i\)
\(L(\frac12)\) \(\approx\) \(0.4128567149 + 0.5325130346i\)
\(L(1)\) \(\approx\) \(0.5760615753 + 0.3741093473i\)
\(L(1)\) \(\approx\) \(0.5760615753 + 0.3741093473i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.391 + 0.920i)T \)
3 \( 1 + (-0.989 + 0.145i)T \)
5 \( 1 + (0.0365 + 0.999i)T \)
7 \( 1 + (0.457 - 0.889i)T \)
11 \( 1 + (0.181 + 0.983i)T \)
13 \( 1 + (0.391 - 0.920i)T \)
17 \( 1 + (0.872 - 0.489i)T \)
19 \( 1 + (-0.639 + 0.768i)T \)
23 \( 1 + (-0.181 + 0.983i)T \)
29 \( 1 + (0.252 + 0.967i)T \)
31 \( 1 + (0.989 + 0.145i)T \)
37 \( 1 + (0.639 - 0.768i)T \)
41 \( 1 + (-0.457 + 0.889i)T \)
43 \( 1 + (-0.694 + 0.719i)T \)
47 \( 1 + (0.833 + 0.551i)T \)
53 \( 1 + (-0.520 - 0.853i)T \)
59 \( 1 + (0.581 + 0.813i)T \)
61 \( 1 + (0.872 + 0.489i)T \)
67 \( 1 + (0.989 - 0.145i)T \)
71 \( 1 + (-0.252 - 0.967i)T \)
73 \( 1 + (0.744 + 0.667i)T \)
79 \( 1 + (-0.833 + 0.551i)T \)
83 \( 1 + (-0.997 - 0.0729i)T \)
89 \( 1 + (0.905 - 0.424i)T \)
97 \( 1 + (0.997 - 0.0729i)T \)
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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

−27.634502019838448425448563795269, −26.66061309527135963451320396193, −25.21575929121508477301501299869, −24.158703765539793147086695111591, −23.39573415987217720301546621332, −21.93425111403526558352758590338, −21.48666118375060039893624719276, −20.637340237082807639049734993410, −19.05173735471758882927817905647, −18.70203748674122228248843551440, −17.31317102302279529115547421917, −16.817819195002404957789069632375, −15.73669887052617348783604895165, −13.875776287014434323953423464053, −12.82582867038594146693630933188, −11.91785040333483996742041769802, −11.38978961473723651162763635080, −10.12345573345861939375669781567, −8.86311318351394459950432145469, −8.168848729769240855496430833165, −6.26090321843601534551063997760, −5.08534108854239744097945010041, −4.09095800660692963284302155950, −2.12491335305989431743243432508, −0.87403487014765048658086565909, 1.29985679850267461108814392643, 3.822003809546992487670565120833, 5.04381082147510555110999365917, 6.16836332545761701087259630818, 7.14317267255948295268999215838, 7.88551236029458014460400229033, 9.90621846463879557822673461468, 10.31526885436634910335986260523, 11.41569810273610438224026155006, 12.92304959274122097466344717357, 14.22552970822690228571477268986, 15.033447945173088742444212610432, 16.04964030784117805901086209470, 17.13682607244407841624950943645, 17.8146975398470621910984206935, 18.46647359641742532123053901750, 19.77793054342916028713078879065, 21.18621819891929239183806429249, 22.50013348192202477104958903623, 23.15447339559094726147965614153, 23.564716910126306671594757276951, 25.06930837122098856594515858040, 25.798926908110850405712118941, 27.107802927466528212864518832766, 27.36225621871776211058047913731

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