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Properties

Label 2-87-29.20-c1-0-5
Degree $2$
Conductor $87$
Sign $0.648 + 0.760i$
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  + (1.78 − 0.859i)2-s + (−0.623 − 0.781i)3-s + (1.19 − 1.50i)4-s + (−1.09 + 0.528i)5-s + (−1.78 − 0.859i)6-s + (0.245 + 0.307i)7-s + (−0.0337 + 0.147i)8-s + (−0.222 + 0.974i)9-s + (−1.50 + 1.88i)10-s + (0.554 + 2.43i)11-s − 1.92·12-s + (−0.735 − 3.22i)13-s + (0.701 + 0.337i)14-s + (1.09 + 0.528i)15-s + (0.922 + 4.04i)16-s − 4.30·17-s + ⋯
L(s)  = 1  + (1.26 − 0.607i)2-s + (−0.359 − 0.451i)3-s + (0.599 − 0.751i)4-s + (−0.491 + 0.236i)5-s + (−0.728 − 0.350i)6-s + (0.0926 + 0.116i)7-s + (−0.0119 + 0.0522i)8-s + (−0.0741 + 0.324i)9-s + (−0.476 + 0.596i)10-s + (0.167 + 0.733i)11-s − 0.555·12-s + (−0.203 − 0.893i)13-s + (0.187 + 0.0903i)14-s + (0.283 + 0.136i)15-s + (0.230 + 1.01i)16-s − 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.648 + 0.760i$
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: \((2,\ 87,\ (\ :1/2),\ 0.648 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31901 - 0.608749i\)
\(L(\frac12)\) \(\approx\) \(1.31901 - 0.608749i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 + 5.05i)T \)
good2 \( 1 + (-1.78 + 0.859i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (1.09 - 0.528i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (-0.245 - 0.307i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-0.554 - 2.43i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.735 + 3.22i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 4.30T + 17T^{2} \)
19 \( 1 + (-3.34 + 4.19i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (2.70 + 1.30i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (-9.01 + 4.34i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (0.812 - 3.55i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + 7.82T + 41T^{2} \)
43 \( 1 + (2.11 + 1.01i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.90 - 8.32i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-9.83 + 4.73i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + (-2.32 - 2.91i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (1.14 - 5.01i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-1.80 - 7.88i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (7.88 + 3.79i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (1.62 - 7.11i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (2.33 - 2.93i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (9.38 - 4.51i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (10.1 - 12.6i)T + (-21.5 - 94.5i)T^{2} \)
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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

−13.63692981013337820844174188839, −13.06333603176451627867364086828, −11.84577863036101994033090550882, −11.46898762746772847264017701163, −10.08377564760850675841894361234, −8.223933758845294494959907695022, −6.85902930013229628673272700306, −5.42641800393009191105941480260, −4.21712110329007367991069909983, −2.54215679615060119698579005786, 3.66740160253107730561586249301, 4.67441406496771689143352280681, 5.90853433365689050427506205142, 7.05101291507953887870061642382, 8.623092770098183662905856474212, 10.08496482764488674377084844213, 11.58396300389461956135187849472, 12.20746569336512079093491929639, 13.61813415860792534176184067111, 14.23330579335308102454371215832

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