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Properties

Label 1-173-173.15-r0-0-0
Degree $1$
Conductor $173$
Sign $0.852 + 0.522i$
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  + (−0.744 + 0.667i)2-s + (−0.639 + 0.768i)3-s + (0.109 − 0.994i)4-s + (0.976 − 0.217i)5-s + (−0.0365 − 0.999i)6-s + (−0.957 + 0.288i)7-s + (0.581 + 0.813i)8-s + (−0.181 − 0.983i)9-s + (−0.581 + 0.813i)10-s + (0.457 − 0.889i)11-s + (0.694 + 0.719i)12-s + (0.744 − 0.667i)13-s + (0.520 − 0.853i)14-s + (−0.457 + 0.889i)15-s + (−0.976 − 0.217i)16-s + (0.997 + 0.0729i)17-s + ⋯
L(s)  = 1  + (−0.744 + 0.667i)2-s + (−0.639 + 0.768i)3-s + (0.109 − 0.994i)4-s + (0.976 − 0.217i)5-s + (−0.0365 − 0.999i)6-s + (−0.957 + 0.288i)7-s + (0.581 + 0.813i)8-s + (−0.181 − 0.983i)9-s + (−0.581 + 0.813i)10-s + (0.457 − 0.889i)11-s + (0.694 + 0.719i)12-s + (0.744 − 0.667i)13-s + (0.520 − 0.853i)14-s + (−0.457 + 0.889i)15-s + (−0.976 − 0.217i)16-s + (0.997 + 0.0729i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.852 + 0.522i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.852 + 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6757651020 + 0.1907402884i\)
\(L(\frac12)\) \(\approx\) \(0.6757651020 + 0.1907402884i\)
\(L(1)\) \(\approx\) \(0.6693291521 + 0.2108431852i\)
\(L(1)\) \(\approx\) \(0.6693291521 + 0.2108431852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.744 + 0.667i)T \)
3 \( 1 + (-0.639 + 0.768i)T \)
5 \( 1 + (0.976 - 0.217i)T \)
7 \( 1 + (-0.957 + 0.288i)T \)
11 \( 1 + (0.457 - 0.889i)T \)
13 \( 1 + (0.744 - 0.667i)T \)
17 \( 1 + (0.997 + 0.0729i)T \)
19 \( 1 + (-0.520 - 0.853i)T \)
23 \( 1 + (-0.457 - 0.889i)T \)
29 \( 1 + (-0.0365 + 0.999i)T \)
31 \( 1 + (0.639 + 0.768i)T \)
37 \( 1 + (0.520 + 0.853i)T \)
41 \( 1 + (0.957 - 0.288i)T \)
43 \( 1 + (0.109 + 0.994i)T \)
47 \( 1 + (-0.934 - 0.357i)T \)
53 \( 1 + (-0.989 + 0.145i)T \)
59 \( 1 + (-0.833 + 0.551i)T \)
61 \( 1 + (0.997 - 0.0729i)T \)
67 \( 1 + (0.639 - 0.768i)T \)
71 \( 1 + (0.0365 - 0.999i)T \)
73 \( 1 + (-0.322 - 0.946i)T \)
79 \( 1 + (0.934 - 0.357i)T \)
83 \( 1 + (0.905 + 0.424i)T \)
89 \( 1 + (-0.872 - 0.489i)T \)
97 \( 1 + (-0.905 + 0.424i)T \)
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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

−27.84291779684702712598196960868, −26.29235517596703492617619052989, −25.52497122813605842123002661926, −24.972050916669273335065457492796, −23.26049987992618480101037928529, −22.5859998278359205674069135036, −21.54245250904533716358540838790, −20.551153659306856018797579980480, −19.25208546364770055102348887287, −18.72592225874746747494311230572, −17.63653181199793018552048796799, −16.996998773929308375642674359641, −16.11640356891191049092899228829, −14.09437375909312261832083034618, −13.12787899291007937725500911123, −12.35426382246860801866177807455, −11.27941673603659231634218149125, −10.0740172388746463908496133232, −9.48782845754394622305145804440, −7.875143980769015773946298093806, −6.7617717235625950666881087060, −5.93915393431385977863318633047, −3.94487164282629204390460392390, −2.33472018912443835038804417278, −1.29900937431769641867258681788, 0.94998176217678905333203303847, 3.1003310866266826181577215168, 4.95742075468153239761190454748, 6.083595951138035078935559153083, 6.42015178241603711100517148807, 8.4903888552949056174078383361, 9.31250323229240353151692020095, 10.18797366467244705896040157648, 11.00089289152094424768388820316, 12.58148953342966607792872027864, 13.89380540109350689385942710925, 15.00317356618074177440713074754, 16.2147185477267216817698217037, 16.54562392844365021579523363469, 17.6367742231811460194395797961, 18.46388049659951266553984087625, 19.666448087467702229281840404838, 20.860018912675922697532859075351, 21.88989868134233050934384834286, 22.75699773990952941130790261144, 23.80151548986333810980267311358, 24.97372524073779405738753338295, 25.779122334291991437174639351722, 26.46977933097645076644866044909, 27.73515301731349307523606786504

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