Properties

Label 2-87-87.86-c6-0-37
Degree $2$
Conductor $87$
Sign $1$
Analytic cond. $20.0147$
Root an. cond. $4.47377$
Motivic weight $6$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13·2-s − 27·3-s + 105·4-s − 351·6-s + 338·7-s + 533·8-s + 729·9-s − 806·11-s − 2.83e3·12-s + 3.00e3·13-s + 4.39e3·14-s + 209·16-s + 9.77e3·17-s + 9.47e3·18-s − 9.12e3·21-s − 1.04e4·22-s − 1.43e4·24-s + 1.56e4·25-s + 3.90e4·26-s − 1.96e4·27-s + 3.54e4·28-s + 2.43e4·29-s − 3.13e4·32-s + 2.17e4·33-s + 1.27e5·34-s + 7.65e4·36-s − 8.10e4·39-s + ⋯
L(s)  = 1  + 13/8·2-s − 3-s + 1.64·4-s − 1.62·6-s + 0.985·7-s + 1.04·8-s + 9-s − 0.605·11-s − 1.64·12-s + 1.36·13-s + 1.60·14-s + 0.0510·16-s + 1.99·17-s + 13/8·18-s − 0.985·21-s − 0.984·22-s − 1.04·24-s + 25-s + 2.22·26-s − 27-s + 1.61·28-s + 29-s − 0.958·32-s + 0.605·33-s + 3.23·34-s + 1.64·36-s − 1.36·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $1$
Analytic conductor: \(20.0147\)
Root analytic conductor: \(4.47377\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: $\chi_{87} (86, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(4.072483265\)
\(L(\frac12)\) \(\approx\) \(4.072483265\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p^{3} T \)
29 \( 1 - p^{3} T \)
good2 \( 1 - 13 T + p^{6} T^{2} \)
5 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
7 \( 1 - 338 T + p^{6} T^{2} \)
11 \( 1 + 806 T + p^{6} T^{2} \)
13 \( 1 - 3002 T + p^{6} T^{2} \)
17 \( 1 - 9778 T + p^{6} T^{2} \)
19 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
23 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
31 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
37 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
41 \( 1 + 132158 T + p^{6} T^{2} \)
43 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
47 \( 1 + 151502 T + p^{6} T^{2} \)
53 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
59 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
61 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
67 \( 1 - 267098 T + p^{6} T^{2} \)
71 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
73 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
79 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
83 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
89 \( 1 - 71266 T + p^{6} T^{2} \)
97 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
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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

−12.95169248687960840627536198520, −12.04015705090309596968891314374, −11.26529281385627449927808417667, −10.31217811105712262840944786851, −8.113935586036240309325670555811, −6.65326012815501190755964108265, −5.52480053080953901258794606163, −4.82322360085907988392024234701, −3.40809270710974888397475475316, −1.34425539106797171657890221211, 1.34425539106797171657890221211, 3.40809270710974888397475475316, 4.82322360085907988392024234701, 5.52480053080953901258794606163, 6.65326012815501190755964108265, 8.113935586036240309325670555811, 10.31217811105712262840944786851, 11.26529281385627449927808417667, 12.04015705090309596968891314374, 12.95169248687960840627536198520

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