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Properties

Label 1-173-173.31-r0-0-0
Degree $1$
Conductor $173$
Sign $0.908 + 0.418i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.908 + 0.418i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.908 + 0.418i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.100592418 + 0.2412415830i\)
\(L(\frac12)\) \(\approx\) \(1.100592418 + 0.2412415830i\)
\(L(1)\) \(\approx\) \(1.042965931 + 0.06021093227i\)
\(L(1)\) \(\approx\) \(1.042965931 + 0.06021093227i\)

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

−27.24738011807263973759204105650, −26.22387348234798368825067809859, −25.48103451271059923599154705654, −25.04211869384046762829289294563, −23.82506915303145680615732416855, −22.99161349446886613660357039030, −21.06681161477183475906799715321, −20.43005476567291984641279393697, −19.74616301690184770936748641466, −18.525206613159744308354582872268, −17.69012160575191022830191857343, −16.56543681776390217889699226026, −15.73331751616230513015390483525, −14.670038525918035369042587731765, −13.47796088161684031595391145758, −12.91583150727204452759196233431, −10.80478571760828672254134382089, −9.776475242585993818923531855679, −9.16946068363629616410703648422, −7.969735991795560780287357257236, −7.13552934674480879677356221262, −5.79236721450870564207464870541, −4.32600260414725865963468510543, −2.513574395842878851332822546826, −1.19051328333794815235867005556, 1.91845831007755517824545395410, 2.742229474203370647461872489265, 3.77640091538982732095369014998, 6.04708696232385548085220145185, 7.153643929424377766791151617365, 8.67989157952106153431099188424, 8.99638190176751128693566775708, 10.34185847562096846223688222731, 11.02946510783198241740182782475, 12.76426290582795400974663891917, 13.37279224642369706525346992048, 14.77721191872231028538891087528, 15.7307960414426006795790150859, 16.834423773691126449866991290233, 18.37560726821162465673065348824, 18.69133193929622297557087853312, 19.52638017519791474564085345775, 20.849061782127289729714160538244, 21.50970971456718199978207478363, 22.15201873866854240898817372771, 24.00936170656211257569531137134, 25.23642662942647863323140397829, 25.89968726850916877051447910663, 26.3066778232871765423268828800, 27.454232896784647679460690847451

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