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Properties

Label 1-173-173.14-r0-0-0
Degree $1$
Conductor $173$
Sign $0.839 - 0.543i$
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  + (−0.934 − 0.357i)2-s + (0.905 + 0.424i)3-s + (0.744 + 0.667i)4-s + (0.109 − 0.994i)5-s + (−0.694 − 0.719i)6-s + (0.989 + 0.145i)7-s + (−0.457 − 0.889i)8-s + (0.639 + 0.768i)9-s + (−0.457 + 0.889i)10-s + (0.520 − 0.853i)11-s + (0.391 + 0.920i)12-s + (−0.934 − 0.357i)13-s + (−0.872 − 0.489i)14-s + (0.520 − 0.853i)15-s + (0.109 + 0.994i)16-s + (−0.0365 − 0.999i)17-s + ⋯
L(s)  = 1  + (−0.934 − 0.357i)2-s + (0.905 + 0.424i)3-s + (0.744 + 0.667i)4-s + (0.109 − 0.994i)5-s + (−0.694 − 0.719i)6-s + (0.989 + 0.145i)7-s + (−0.457 − 0.889i)8-s + (0.639 + 0.768i)9-s + (−0.457 + 0.889i)10-s + (0.520 − 0.853i)11-s + (0.391 + 0.920i)12-s + (−0.934 − 0.357i)13-s + (−0.872 − 0.489i)14-s + (0.520 − 0.853i)15-s + (0.109 + 0.994i)16-s + (−0.0365 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.839 - 0.543i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.073765854 - 0.3169267141i\)
\(L(\frac12)\) \(\approx\) \(1.073765854 - 0.3169267141i\)
\(L(1)\) \(\approx\) \(1.014736750 - 0.1821929477i\)
\(L(1)\) \(\approx\) \(1.014736750 - 0.1821929477i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.934 - 0.357i)T \)
3 \( 1 + (0.905 + 0.424i)T \)
5 \( 1 + (0.109 - 0.994i)T \)
7 \( 1 + (0.989 + 0.145i)T \)
11 \( 1 + (0.520 - 0.853i)T \)
13 \( 1 + (-0.934 - 0.357i)T \)
17 \( 1 + (-0.0365 - 0.999i)T \)
19 \( 1 + (-0.872 + 0.489i)T \)
23 \( 1 + (0.520 + 0.853i)T \)
29 \( 1 + (-0.694 + 0.719i)T \)
31 \( 1 + (0.905 - 0.424i)T \)
37 \( 1 + (-0.872 + 0.489i)T \)
41 \( 1 + (0.989 + 0.145i)T \)
43 \( 1 + (0.744 - 0.667i)T \)
47 \( 1 + (-0.181 - 0.983i)T \)
53 \( 1 + (-0.997 - 0.0729i)T \)
59 \( 1 + (0.957 + 0.288i)T \)
61 \( 1 + (-0.0365 + 0.999i)T \)
67 \( 1 + (0.905 + 0.424i)T \)
71 \( 1 + (-0.694 + 0.719i)T \)
73 \( 1 + (-0.581 - 0.813i)T \)
79 \( 1 + (-0.181 + 0.983i)T \)
83 \( 1 + (-0.976 + 0.217i)T \)
89 \( 1 + (0.252 + 0.967i)T \)
97 \( 1 + (-0.976 - 0.217i)T \)
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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

−27.27184181913310885295760224156, −26.45448696337928225092901639048, −25.883981825966639635625281358861, −24.8074328084943064802645030565, −24.17976780699505237343214761075, −23.0366391508836304187561995737, −21.47552653299846581614062697034, −20.56244348547137322645254635634, −19.3780704008136632093532066608, −18.99746638922109253851174583630, −17.614650454764159682430752012268, −17.36906867056671087717969222886, −15.429394992076656428198031668280, −14.64516936872778749305619842250, −14.304342627366265275516248073747, −12.5116088479564117830199437913, −11.21422360591992027070500532558, −10.18662750175946659331742785234, −9.153274707181696446171319834752, −8.02555632224962470382321994362, −7.19931835105016412221482842781, −6.37853405421362535722027206857, −4.39069140132646770241336598694, −2.52266701678701534662336658654, −1.73173807851816686335261864840, 1.33363151263977587944482434726, 2.55438092630868643902372849205, 4.01042916634488830497586997865, 5.31255615614524911805439422126, 7.34557542149862954222867268998, 8.32179253128400301539602950585, 8.96621708485764665238152858062, 9.89418849351226493842487376762, 11.15464152434361218317401812802, 12.20056691564425581480280580307, 13.42712381553309675361997465465, 14.6380627741555043386314901144, 15.730819069429031206071970593860, 16.73928726808991366899813896676, 17.50883396019418771831377002503, 18.881585827739168094567331198, 19.65438800203945498660040822367, 20.634959382558388073239265001895, 21.12143365054414405245575489652, 22.05369560920246963697979068226, 24.173714053627988380876648941475, 24.75295073288096623296157438781, 25.44136946171893387065487352294, 26.77484915287163810598076850754, 27.44800987920679798100109308662

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