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Properties

Label 1-173-173.38-r0-0-0
Degree $1$
Conductor $173$
Sign $0.645 - 0.763i$
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  + (0.322 + 0.946i)2-s + (−0.520 + 0.853i)3-s + (−0.791 + 0.611i)4-s + (−0.252 − 0.967i)5-s + (−0.976 − 0.217i)6-s + (0.181 − 0.983i)7-s + (−0.833 − 0.551i)8-s + (−0.457 − 0.889i)9-s + (0.833 − 0.551i)10-s + (−0.957 − 0.288i)11-s + (−0.109 − 0.994i)12-s + (−0.322 − 0.946i)13-s + (0.989 − 0.145i)14-s + (0.957 + 0.288i)15-s + (0.252 − 0.967i)16-s + (−0.905 + 0.424i)17-s + ⋯
L(s)  = 1  + (0.322 + 0.946i)2-s + (−0.520 + 0.853i)3-s + (−0.791 + 0.611i)4-s + (−0.252 − 0.967i)5-s + (−0.976 − 0.217i)6-s + (0.181 − 0.983i)7-s + (−0.833 − 0.551i)8-s + (−0.457 − 0.889i)9-s + (0.833 − 0.551i)10-s + (−0.957 − 0.288i)11-s + (−0.109 − 0.994i)12-s + (−0.322 − 0.946i)13-s + (0.989 − 0.145i)14-s + (0.957 + 0.288i)15-s + (0.252 − 0.967i)16-s + (−0.905 + 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.645 - 0.763i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.645 - 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4153554514 - 0.1927453939i\)
\(L(\frac12)\) \(\approx\) \(0.4153554514 - 0.1927453939i\)
\(L(1)\) \(\approx\) \(0.6564714495 + 0.1964839342i\)
\(L(1)\) \(\approx\) \(0.6564714495 + 0.1964839342i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.322 + 0.946i)T \)
3 \( 1 + (-0.520 + 0.853i)T \)
5 \( 1 + (-0.252 - 0.967i)T \)
7 \( 1 + (0.181 - 0.983i)T \)
11 \( 1 + (-0.957 - 0.288i)T \)
13 \( 1 + (-0.322 - 0.946i)T \)
17 \( 1 + (-0.905 + 0.424i)T \)
19 \( 1 + (-0.989 - 0.145i)T \)
23 \( 1 + (0.957 - 0.288i)T \)
29 \( 1 + (-0.976 + 0.217i)T \)
31 \( 1 + (0.520 + 0.853i)T \)
37 \( 1 + (0.989 + 0.145i)T \)
41 \( 1 + (-0.181 + 0.983i)T \)
43 \( 1 + (-0.791 - 0.611i)T \)
47 \( 1 + (-0.581 - 0.813i)T \)
53 \( 1 + (-0.639 - 0.768i)T \)
59 \( 1 + (0.934 + 0.357i)T \)
61 \( 1 + (-0.905 - 0.424i)T \)
67 \( 1 + (0.520 - 0.853i)T \)
71 \( 1 + (0.976 - 0.217i)T \)
73 \( 1 + (0.391 - 0.920i)T \)
79 \( 1 + (0.581 - 0.813i)T \)
83 \( 1 + (-0.872 - 0.489i)T \)
89 \( 1 + (-0.997 - 0.0729i)T \)
97 \( 1 + (0.872 - 0.489i)T \)
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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

−27.97618573277360680732663311855, −26.86951972976592611509152576759, −25.73471723750860090037987000808, −24.40893014756681572502140884198, −23.55423903676554315366850380840, −22.7179822364190513770575328422, −21.9386303678154733540056834042, −21.029807827232298535984715963705, −19.54969159600155224728447760806, −18.74780637269807688760360778634, −18.3537910925366961993640525544, −17.27376830241047792483801413252, −15.47252249951500892227959982851, −14.5729500688084296127337140997, −13.38807735099059633187688650952, −12.53905654819944855324712898633, −11.416690215791185665793619315642, −11.02955643930324166033842663974, −9.573055906010595273112687463997, −8.19353435358300269776213748556, −6.827176190731607358083547829979, −5.72719346565696195393375242385, −4.52189700854002196146694449504, −2.66227111070346464365906161035, −2.06646122838920086578699169329, 0.34095594766002941533772151291, 3.4471485772325665339386342752, 4.62684395593507778395544372030, 5.1538614706417619929218784621, 6.502987001496917610731645020926, 7.9172197099156112608305826126, 8.77521854828561862128446336802, 10.091505244816343193660630235170, 11.16347624428248230474795912153, 12.723653136790839672658079536174, 13.30244317679475254417607061673, 14.8773047955394975552310485282, 15.54609434479561417894079164280, 16.60827905816111243225909881444, 17.08674556406740459229660679452, 18.034949889331428160932778916665, 19.84984121898597417330947524121, 20.84092398631638631254600608784, 21.6079428222726400366633816092, 22.848285070764272022937254227654, 23.51888100985443833595879150251, 24.22197818682838151726830496991, 25.4309638285634522315958206306, 26.61295513407452986343795258114, 27.07674087604998657022660447583

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