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Properties

Label 2-87-87.86-c4-0-25
Degree $2$
Conductor $87$
Sign $-0.0149 + 0.999i$
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  − 3.96·2-s + (6.34 − 6.38i)3-s − 0.286·4-s + 7.48i·5-s + (−25.1 + 25.3i)6-s + 18.5·7-s + 64.5·8-s + (−0.485 − 80.9i)9-s − 29.6i·10-s + 141.·11-s + (−1.81 + 1.82i)12-s − 227.·13-s − 73.5·14-s + (47.7 + 47.4i)15-s − 251.·16-s + 50.8·17-s + ⋯
L(s)  = 1  − 0.991·2-s + (0.704 − 0.709i)3-s − 0.0178·4-s + 0.299i·5-s + (−0.698 + 0.702i)6-s + 0.378·7-s + 1.00·8-s + (−0.00599 − 0.999i)9-s − 0.296i·10-s + 1.17·11-s + (−0.0126 + 0.0126i)12-s − 1.34·13-s − 0.375·14-s + (0.212 + 0.211i)15-s − 0.981·16-s + 0.175·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0149 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0149 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.0149 + 0.999i$
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: \((2,\ 87,\ (\ :2),\ -0.0149 + 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.766905 - 0.778481i\)
\(L(\frac12)\) \(\approx\) \(0.766905 - 0.778481i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-6.34 + 6.38i)T \)
29 \( 1 + (605. - 583. i)T \)
good2 \( 1 + 3.96T + 16T^{2} \)
5 \( 1 - 7.48iT - 625T^{2} \)
7 \( 1 - 18.5T + 2.40e3T^{2} \)
11 \( 1 - 141.T + 1.46e4T^{2} \)
13 \( 1 + 227.T + 2.85e4T^{2} \)
17 \( 1 - 50.8T + 8.35e4T^{2} \)
19 \( 1 + 628. iT - 1.30e5T^{2} \)
23 \( 1 + 898. iT - 2.79e5T^{2} \)
31 \( 1 + 1.34e3iT - 9.23e5T^{2} \)
37 \( 1 - 550. iT - 1.87e6T^{2} \)
41 \( 1 - 1.77e3T + 2.82e6T^{2} \)
43 \( 1 + 916. iT - 3.41e6T^{2} \)
47 \( 1 - 1.68e3T + 4.87e6T^{2} \)
53 \( 1 - 2.49e3iT - 7.89e6T^{2} \)
59 \( 1 - 3.14e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.51e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.13e3T + 2.01e7T^{2} \)
71 \( 1 + 4.51e3iT - 2.54e7T^{2} \)
73 \( 1 + 7.19e3iT - 2.83e7T^{2} \)
79 \( 1 - 1.90e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.66e3iT - 4.74e7T^{2} \)
89 \( 1 - 5.57e3T + 6.27e7T^{2} \)
97 \( 1 - 1.44e4iT - 8.85e7T^{2} \)
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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

−13.24786850242319575694456984996, −12.11184616273662140439477228103, −10.82613508653351068030724258775, −9.428365263978210943509995560776, −8.840794383537050967454333616921, −7.57935180561875405897836261102, −6.76984943762445415721924994130, −4.49000379814721588068824052075, −2.42296424671256603128031675873, −0.74029785998746121567774473702, 1.62792745716942484322292906249, 3.84456513000084003501745872549, 5.13609939593843486810135380650, 7.41789893934898326525889757280, 8.327222616116243687528679303886, 9.424675563126982685236926319135, 9.910087938211625937350946575562, 11.23890706051577095632993634341, 12.62913123431535449921776585573, 14.16274674862484012017203044884

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