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Properties

Label 2-87-87.86-c4-0-2
Degree $2$
Conductor $87$
Sign $-0.856 - 0.516i$
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  + 0.630·2-s + (7.48 − 5.00i)3-s − 15.6·4-s + 33.7i·5-s + (4.71 − 3.15i)6-s − 48.1·7-s − 19.9·8-s + (30.9 − 74.8i)9-s + 21.2i·10-s − 157.·11-s + (−116. + 78.0i)12-s − 194.·13-s − 30.3·14-s + (168. + 252. i)15-s + 237.·16-s + 111.·17-s + ⋯
L(s)  = 1  + 0.157·2-s + (0.831 − 0.555i)3-s − 0.975·4-s + 1.35i·5-s + (0.130 − 0.0875i)6-s − 0.982·7-s − 0.311·8-s + (0.382 − 0.923i)9-s + 0.212i·10-s − 1.30·11-s + (−0.810 + 0.541i)12-s − 1.15·13-s − 0.154·14-s + (0.750 + 1.12i)15-s + 0.926·16-s + 0.385·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.856 - 0.516i$
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: \((2,\ 87,\ (\ :2),\ -0.856 - 0.516i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.130693 + 0.469490i\)
\(L(\frac12)\) \(\approx\) \(0.130693 + 0.469490i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-7.48 + 5.00i)T \)
29 \( 1 + (357. + 761. i)T \)
good2 \( 1 - 0.630T + 16T^{2} \)
5 \( 1 - 33.7iT - 625T^{2} \)
7 \( 1 + 48.1T + 2.40e3T^{2} \)
11 \( 1 + 157.T + 1.46e4T^{2} \)
13 \( 1 + 194.T + 2.85e4T^{2} \)
17 \( 1 - 111.T + 8.35e4T^{2} \)
19 \( 1 - 542. iT - 1.30e5T^{2} \)
23 \( 1 - 419. iT - 2.79e5T^{2} \)
31 \( 1 + 340. iT - 9.23e5T^{2} \)
37 \( 1 - 1.47e3iT - 1.87e6T^{2} \)
41 \( 1 + 205.T + 2.82e6T^{2} \)
43 \( 1 + 2.13e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.20e3T + 4.87e6T^{2} \)
53 \( 1 + 1.41e3iT - 7.89e6T^{2} \)
59 \( 1 - 44.2iT - 1.21e7T^{2} \)
61 \( 1 - 1.30e3iT - 1.38e7T^{2} \)
67 \( 1 + 3.05e3T + 2.01e7T^{2} \)
71 \( 1 - 350. iT - 2.54e7T^{2} \)
73 \( 1 - 1.02e4iT - 2.83e7T^{2} \)
79 \( 1 - 4.69e3iT - 3.89e7T^{2} \)
83 \( 1 + 3.22e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.27e4T + 6.27e7T^{2} \)
97 \( 1 - 1.40e4iT - 8.85e7T^{2} \)
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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

−13.87417582029706586119179837807, −13.06594985822278322932352222107, −12.13130835946120674050276692591, −10.15277051887830094034451547190, −9.733406783806592262735966104484, −8.088983919620954062428541323965, −7.20520310962558023364069984679, −5.76809357681251705146727469410, −3.66410181684537829481014770412, −2.65323429661869584328951826225, 0.19702113337671870745287815673, 2.90890250406956142528236913434, 4.55045047656397811035514467725, 5.24295323632719875921713165393, 7.62464465515526434267242932334, 8.834202462877346406118125656656, 9.398079086071637546493500364797, 10.39913684666297446942191356478, 12.67379129372916916849598274560, 12.88033311454821478588737930923

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