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Properties

Label 2-87-87.86-c4-0-31
Degree $2$
Conductor $87$
Sign $0.411 + 0.911i$
Analytic cond. $8.99318$
Root an. cond. $2.99886$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.29·2-s + (7.78 − 4.52i)3-s + 2.41·4-s − 28.6i·5-s + (33.3 − 19.4i)6-s + 23.9·7-s − 58.3·8-s + (40.0 − 70.3i)9-s − 122. i·10-s + 59.0·11-s + (18.7 − 10.9i)12-s − 28.6·13-s + 102.·14-s + (−129. − 222. i)15-s − 288.·16-s + 333.·17-s + ⋯
L(s)  = 1  + 1.07·2-s + (0.864 − 0.502i)3-s + 0.150·4-s − 1.14i·5-s + (0.927 − 0.539i)6-s + 0.488·7-s − 0.911·8-s + (0.494 − 0.869i)9-s − 1.22i·10-s + 0.488·11-s + (0.130 − 0.0757i)12-s − 0.169·13-s + 0.524·14-s + (−0.576 − 0.990i)15-s − 1.12·16-s + 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.411 + 0.911i$
Analytic conductor: \(8.99318\)
Root analytic conductor: \(2.99886\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :2),\ 0.411 + 0.911i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.92730 - 1.89046i\)
\(L(\frac12)\) \(\approx\) \(2.92730 - 1.89046i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-7.78 + 4.52i)T \)
29 \( 1 + (86.2 - 836. i)T \)
good2 \( 1 - 4.29T + 16T^{2} \)
5 \( 1 + 28.6iT - 625T^{2} \)
7 \( 1 - 23.9T + 2.40e3T^{2} \)
11 \( 1 - 59.0T + 1.46e4T^{2} \)
13 \( 1 + 28.6T + 2.85e4T^{2} \)
17 \( 1 - 333.T + 8.35e4T^{2} \)
19 \( 1 - 184. iT - 1.30e5T^{2} \)
23 \( 1 - 632. iT - 2.79e5T^{2} \)
31 \( 1 + 461. iT - 9.23e5T^{2} \)
37 \( 1 - 351. iT - 1.87e6T^{2} \)
41 \( 1 - 2.02e3T + 2.82e6T^{2} \)
43 \( 1 - 26.2iT - 3.41e6T^{2} \)
47 \( 1 + 2.57e3T + 4.87e6T^{2} \)
53 \( 1 - 4.05e3iT - 7.89e6T^{2} \)
59 \( 1 - 986. iT - 1.21e7T^{2} \)
61 \( 1 + 1.53e3iT - 1.38e7T^{2} \)
67 \( 1 + 5.18e3T + 2.01e7T^{2} \)
71 \( 1 - 5.46e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.75e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.15e4iT - 3.89e7T^{2} \)
83 \( 1 + 3.85e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.21e4T + 6.27e7T^{2} \)
97 \( 1 + 1.58e4iT - 8.85e7T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.25463304022903861757078736930, −12.51555721408156059033937835747, −11.76400439385550262252708926964, −9.599668632353052455023662744701, −8.737807527153742048061063328015, −7.60716652792785806242942381613, −5.85761839672619828549037745304, −4.62187720573628618918296898120, −3.37200101194967065245811054433, −1.35004408287197543257833754279, 2.61796766598672772398054164716, 3.70709292241022004751250826642, 4.93355534104630732696753914624, 6.51706166900226452249505869095, 7.938698172589534650373714203380, 9.282904648291576728541722843639, 10.43497262884173892521142432591, 11.60549776554343126553869956031, 12.88219449112230559026502300820, 14.05054969118010333908276524928

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