↑

Properties

Label 2-87-3.2-c2-0-10
Degree $2$
Conductor $87$
Sign $0.330 + 0.943i$
Analytic cond. $2.37057$
Root an. cond. $1.53966$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.949i·2-s + (−2.83 + 0.992i)3-s + 3.09·4-s − 4.66i·5-s + (0.941 + 2.68i)6-s + 2.70·7-s − 6.73i·8-s + (7.02 − 5.61i)9-s − 4.43·10-s − 13.8i·11-s + (−8.77 + 3.07i)12-s + 6.42·13-s − 2.57i·14-s + (4.63 + 13.2i)15-s + 6.00·16-s + 8.06i·17-s + ⋯
L(s)  = 1  − 0.474i·2-s + (−0.943 + 0.330i)3-s + 0.774·4-s − 0.933i·5-s + (0.156 + 0.447i)6-s + 0.386·7-s − 0.842i·8-s + (0.781 − 0.624i)9-s − 0.443·10-s − 1.25i·11-s + (−0.731 + 0.256i)12-s + 0.494·13-s − 0.183i·14-s + (0.308 + 0.881i)15-s + 0.375·16-s + 0.474i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.330 + 0.943i$
Analytic conductor: \(2.37057\)
Root analytic conductor: \(1.53966\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1),\ 0.330 + 0.943i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.977322 - 0.693003i\)
\(L(\frac12)\) \(\approx\) \(0.977322 - 0.693003i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.83 - 0.992i)T \)
29 \( 1 - 5.38iT \)
good2 \( 1 + 0.949iT - 4T^{2} \)
5 \( 1 + 4.66iT - 25T^{2} \)
7 \( 1 - 2.70T + 49T^{2} \)
11 \( 1 + 13.8iT - 121T^{2} \)
13 \( 1 - 6.42T + 169T^{2} \)
17 \( 1 - 8.06iT - 289T^{2} \)
19 \( 1 + 18.9T + 361T^{2} \)
23 \( 1 - 45.5iT - 529T^{2} \)
31 \( 1 - 0.734T + 961T^{2} \)
37 \( 1 - 45.0T + 1.36e3T^{2} \)
41 \( 1 + 22.1iT - 1.68e3T^{2} \)
43 \( 1 + 54.1T + 1.84e3T^{2} \)
47 \( 1 - 52.7iT - 2.20e3T^{2} \)
53 \( 1 + 10.8iT - 2.80e3T^{2} \)
59 \( 1 - 0.612iT - 3.48e3T^{2} \)
61 \( 1 - 24.1T + 3.72e3T^{2} \)
67 \( 1 - 106.T + 4.48e3T^{2} \)
71 \( 1 - 25.3iT - 5.04e3T^{2} \)
73 \( 1 + 18.3T + 5.32e3T^{2} \)
79 \( 1 - 141.T + 6.24e3T^{2} \)
83 \( 1 - 155. iT - 6.88e3T^{2} \)
89 \( 1 - 63.6iT - 7.92e3T^{2} \)
97 \( 1 - 12.1T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.33420509583491953127028665616, −12.45251569435225311260816758473, −11.35707771654618434107339646042, −10.91403548978543741234254080991, −9.546062319420633622401233363054, −8.154155473673289968346958606204, −6.45736227613958168745607102872, −5.39709796074698613578257019277, −3.77005744218312804810104770247, −1.23626821902947062873504699474, 2.23094328715608785118440747255, 4.76647015355168225410444354616, 6.37883110819705734276190491718, 6.89015893019170159962266260511, 8.077912089395606928099668937046, 10.22088861152857112265779065369, 10.95653404611848416991445794001, 11.85941917778849530823045687083, 12.91988011438365368158616996240, 14.54011924591986649091626753785

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