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Properties

Label 2-87-29.23-c1-0-1
Degree $2$
Conductor $87$
Sign $-0.243 - 0.969i$
Analytic cond. $0.694698$
Root an. cond. $0.833485$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.754 + 0.946i)2-s + (−0.222 + 0.974i)3-s + (0.118 + 0.520i)4-s + (1.12 − 1.40i)5-s + (−0.754 − 0.946i)6-s + (−0.951 + 4.16i)7-s + (−2.76 − 1.33i)8-s + (−0.900 − 0.433i)9-s + (0.484 + 2.12i)10-s + (−0.951 + 0.458i)11-s − 0.534·12-s + (4.75 − 2.29i)13-s + (−3.22 − 4.04i)14-s + (1.12 + 1.40i)15-s + (2.38 − 1.14i)16-s + 5.61·17-s + ⋯
L(s)  = 1  + (−0.533 + 0.669i)2-s + (−0.128 + 0.562i)3-s + (0.0594 + 0.260i)4-s + (0.501 − 0.628i)5-s + (−0.308 − 0.386i)6-s + (−0.359 + 1.57i)7-s + (−0.977 − 0.470i)8-s + (−0.300 − 0.144i)9-s + (0.153 + 0.671i)10-s + (−0.286 + 0.138i)11-s − 0.154·12-s + (1.31 − 0.635i)13-s + (−0.862 − 1.08i)14-s + (0.289 + 0.363i)15-s + (0.596 − 0.287i)16-s + 1.36·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.243 - 0.969i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ -0.243 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.469740 + 0.602017i\)
\(L(\frac12)\) \(\approx\) \(0.469740 + 0.602017i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.222 - 0.974i)T \)
29 \( 1 + (4.54 + 2.88i)T \)
good2 \( 1 + (0.754 - 0.946i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (-1.12 + 1.40i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.951 - 4.16i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (0.951 - 0.458i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-4.75 + 2.29i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 - 5.61T + 17T^{2} \)
19 \( 1 + (1.42 + 6.25i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (-1.27 - 1.59i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (-2.38 + 2.99i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (-0.315 - 0.151i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + 3.62T + 41T^{2} \)
43 \( 1 + (-1.09 - 1.36i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (6.28 - 3.02i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (4.87 - 6.11i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 - 0.382T + 59T^{2} \)
61 \( 1 + (-1.21 + 5.32i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (7.03 + 3.38i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-14.0 + 6.77i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (1.58 + 1.98i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (9.25 + 4.45i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (0.338 + 1.48i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (11.1 - 14.0i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-1.51 - 6.63i)T + (-87.3 + 42.0i)T^{2} \)
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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

−15.15152678896365793711870784157, −13.26551628657602676815675150162, −12.45953994059274232264798239126, −11.29993887468933255628308287625, −9.599277752725495073937866476344, −8.987138622015171827050720277254, −8.025347518846611524254159839140, −6.19504583360263124268831505385, −5.36399750697923939943838110292, −3.11329208469633293741539300682, 1.40518222025409769858113908078, 3.45775692869429023077446155565, 5.90358875668298089693931688396, 6.88335105188030357855003100319, 8.332294506402500083140509045419, 9.981543815074833337381663571605, 10.47934467393749412346609923507, 11.42246077178445240105968492306, 12.79094537909157446305172740334, 13.98071191794682076709509223026

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