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Properties

Label 2-87-29.20-c1-0-0
Degree $2$
Conductor $87$
Sign $-0.431 - 0.902i$
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  + (−1.04 + 0.505i)2-s + (0.623 + 0.781i)3-s + (−0.402 + 0.504i)4-s + (−2.80 + 1.34i)5-s + (−1.04 − 0.505i)6-s + (1.08 + 1.36i)7-s + (0.685 − 3.00i)8-s + (−0.222 + 0.974i)9-s + (2.25 − 2.83i)10-s + (0.616 + 2.70i)11-s − 0.645·12-s + (0.256 + 1.12i)13-s + (−1.83 − 0.881i)14-s + (−2.80 − 1.34i)15-s + (0.510 + 2.23i)16-s + 2.64·17-s + ⋯
L(s)  = 1  + (−0.741 + 0.357i)2-s + (0.359 + 0.451i)3-s + (−0.201 + 0.252i)4-s + (−1.25 + 0.603i)5-s + (−0.428 − 0.206i)6-s + (0.411 + 0.516i)7-s + (0.242 − 1.06i)8-s + (−0.0741 + 0.324i)9-s + (0.713 − 0.895i)10-s + (0.185 + 0.814i)11-s − 0.186·12-s + (0.0711 + 0.311i)13-s + (−0.489 − 0.235i)14-s + (−0.723 − 0.348i)15-s + (0.127 + 0.558i)16-s + 0.640·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.431 - 0.902i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ -0.431 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.317951 + 0.504298i\)
\(L(\frac12)\) \(\approx\) \(0.317951 + 0.504298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (0.497 + 5.36i)T \)
good2 \( 1 + (1.04 - 0.505i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (2.80 - 1.34i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (-1.08 - 1.36i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-0.616 - 2.70i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.256 - 1.12i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 - 2.64T + 17T^{2} \)
19 \( 1 + (-4.50 + 5.64i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (-2.53 - 1.21i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (3.59 - 1.72i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (2.32 - 10.2i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 - 8.92T + 41T^{2} \)
43 \( 1 + (10.6 + 5.11i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.63 - 7.16i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-4.85 + 2.33i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + 8.20T + 59T^{2} \)
61 \( 1 + (-3.06 - 3.83i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (-0.100 + 0.442i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (2.58 + 11.3i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-11.5 - 5.57i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (-3.43 + 15.0i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (2.62 - 3.29i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-1.83 + 0.881i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (-7.89 + 9.90i)T + (-21.5 - 94.5i)T^{2} \)
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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

−15.03250383003224433475706340885, −13.62629313164745533669374177772, −12.16412319801309810542489344732, −11.30253711149472091239664469975, −9.873144917354784594129402167994, −8.892401851231978493002805853138, −7.81212171348799932359362160075, −7.06024149123853807877717512999, −4.68568179060436939562462657441, −3.33691846232215542560157743956, 1.03469816379415901820362250377, 3.70638328448883270835269901696, 5.40062738347350594289857587293, 7.57600690310369568325346877976, 8.216596832340254768817281801951, 9.226443399594038151507074584473, 10.64711004056893654702777064663, 11.57750000922513715609162045384, 12.61046406352369509101433855644, 13.99071966141046188381665685582

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