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Properties

Label 1-87-87.65-r1-0-0
Degree $1$
Conductor $87$
Sign $0.620 - 0.784i$
Analytic cond. $9.34944$
Root an. cond. $9.34944$
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  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (0.900 − 0.433i)5-s + (0.623 + 0.781i)7-s + (0.222 − 0.974i)8-s + (0.623 − 0.781i)10-s + (0.222 + 0.974i)11-s + (−0.222 − 0.974i)13-s + (0.900 + 0.433i)14-s + (−0.222 − 0.974i)16-s − 17-s + (0.623 − 0.781i)19-s + (0.222 − 0.974i)20-s + (0.623 + 0.781i)22-s + (0.900 + 0.433i)23-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (0.900 − 0.433i)5-s + (0.623 + 0.781i)7-s + (0.222 − 0.974i)8-s + (0.623 − 0.781i)10-s + (0.222 + 0.974i)11-s + (−0.222 − 0.974i)13-s + (0.900 + 0.433i)14-s + (−0.222 − 0.974i)16-s − 17-s + (0.623 − 0.781i)19-s + (0.222 − 0.974i)20-s + (0.623 + 0.781i)22-s + (0.900 + 0.433i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.620 - 0.784i$
Analytic conductor: \(9.34944\)
Root analytic conductor: \(9.34944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (1:\ ),\ 0.620 - 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.074473199 - 1.487943756i\)
\(L(\frac12)\) \(\approx\) \(3.074473199 - 1.487943756i\)
\(L(1)\) \(\approx\) \(2.034796778 - 0.6646398613i\)
\(L(1)\) \(\approx\) \(2.034796778 - 0.6646398613i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 - T \)
19 \( 1 + (0.623 - 0.781i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 - T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 + (-0.222 + 0.974i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−30.61174703075594964188419402777, −29.58812434467475591563239532367, −28.96268183345549086381135593168, −26.83871863290863781403161139776, −26.3578280150771836173929411538, −24.916240657713525608381057752743, −24.2507474223489320804056760911, −23.09077724575082862192277179073, −21.93422905196516837351733368782, −21.24353703817588236675598179532, −20.133765069368202563629090684105, −18.466359295009783274517329561083, −17.15664714265467618308907718499, −16.45867057415273744318602260576, −14.801349294109510922093423793795, −13.99840541337332123735435758354, −13.27842099991435822110138494050, −11.59320072453452693143385743307, −10.6572152395677082342251983488, −8.87425705830416983144927078231, −7.29781406343544425197566443842, −6.29367595492630575307423519530, −4.98207692120370838368163502780, −3.56896107528773661768421297625, −1.91157241254644687255053216904, 1.53757036526666219910310795667, 2.736190153321115035996713277519, 4.76454107133526592269711135705, 5.48093594988323845199846262105, 6.97144020924810796305212671097, 8.93702979646417961799369655832, 10.11293865907914517906080711511, 11.45625860883127710267248802645, 12.61269445465736813154551570058, 13.46608526410051180619829467729, 14.793546674784794224923536905655, 15.57077619506746820331330479718, 17.33355102440142441434300083532, 18.26050408736579818811314223531, 19.93023483697598364094713332252, 20.65281571395242262904441261814, 21.78631146472896095849824488080, 22.43469445421227648074507252001, 23.90445874369800443501748698979, 24.85384840207283309324941766439, 25.47987563813625853236517551845, 27.455376416853339195155964066931, 28.432051226185119082541591581039, 29.16395109764679695426643705373, 30.43216282527561611863171786422

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