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Properties

Label 2-87-87.86-c4-0-27
Degree $2$
Conductor $87$
Sign $0.982 + 0.188i$
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  + 6.51·2-s + (5.97 − 6.72i)3-s + 26.5·4-s + 31.1i·5-s + (38.9 − 43.8i)6-s + 44.9·7-s + 68.5·8-s + (−9.48 − 80.4i)9-s + 202. i·10-s − 146.·11-s + (158. − 178. i)12-s + 138.·13-s + 292.·14-s + (209. + 186. i)15-s + 22.5·16-s − 370.·17-s + ⋯
L(s)  = 1  + 1.62·2-s + (0.664 − 0.747i)3-s + 1.65·4-s + 1.24i·5-s + (1.08 − 1.21i)6-s + 0.916·7-s + 1.07·8-s + (−0.117 − 0.993i)9-s + 2.02i·10-s − 1.20·11-s + (1.10 − 1.23i)12-s + 0.817·13-s + 1.49·14-s + (0.930 + 0.827i)15-s + 0.0881·16-s − 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.982 + 0.188i$
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: \((2,\ 87,\ (\ :2),\ 0.982 + 0.188i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(4.83999 - 0.459317i\)
\(L(\frac12)\) \(\approx\) \(4.83999 - 0.459317i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.97 + 6.72i)T \)
29 \( 1 + (-430. - 722. i)T \)
good2 \( 1 - 6.51T + 16T^{2} \)
5 \( 1 - 31.1iT - 625T^{2} \)
7 \( 1 - 44.9T + 2.40e3T^{2} \)
11 \( 1 + 146.T + 1.46e4T^{2} \)
13 \( 1 - 138.T + 2.85e4T^{2} \)
17 \( 1 + 370.T + 8.35e4T^{2} \)
19 \( 1 + 506. iT - 1.30e5T^{2} \)
23 \( 1 - 46.4iT - 2.79e5T^{2} \)
31 \( 1 - 807. iT - 9.23e5T^{2} \)
37 \( 1 - 409. iT - 1.87e6T^{2} \)
41 \( 1 + 897.T + 2.82e6T^{2} \)
43 \( 1 - 1.57e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.76e3T + 4.87e6T^{2} \)
53 \( 1 + 3.66e3iT - 7.89e6T^{2} \)
59 \( 1 - 5.90e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.06e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.36e3T + 2.01e7T^{2} \)
71 \( 1 + 4.09e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.77e3iT - 2.83e7T^{2} \)
79 \( 1 - 6.09e3iT - 3.89e7T^{2} \)
83 \( 1 - 4.86e3iT - 4.74e7T^{2} \)
89 \( 1 - 2.04e3T + 6.27e7T^{2} \)
97 \( 1 + 6.05e3iT - 8.85e7T^{2} \)
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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

−13.59068705493836390369259329547, −12.79588877839280030670115900589, −11.40922160367410428393578144428, −10.80796512250698610185872284734, −8.606772692944831864383666904320, −7.21553983771671751751301970974, −6.42875374565904602219644270277, −4.86810741516807410984405781083, −3.22734733111910007296611151491, −2.28127644204221460830365524703, 2.19664767738612109591243004818, 3.97996734476891377192402612483, 4.79346382920342195234684942168, 5.71743909540919152238922417885, 7.924904245966939029317543745653, 8.825013441786189126157082432461, 10.50838913064069600086733145275, 11.58794314933490047523094651608, 12.84354345443405780211975473013, 13.50582596196196198051312752996

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