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Properties

Label 2-87-29.4-c1-0-1
Degree $2$
Conductor $87$
Sign $0.893 - 0.449i$
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  + (1.26 + 0.289i)2-s + (0.433 + 0.900i)3-s + (−0.274 − 0.132i)4-s + (−0.0725 + 0.317i)5-s + (0.289 + 1.26i)6-s + (0.994 − 0.478i)7-s + (−2.34 − 1.87i)8-s + (−0.623 + 0.781i)9-s + (−0.184 + 0.382i)10-s + (−0.600 + 0.478i)11-s − 0.304i·12-s + (−2.99 − 3.76i)13-s + (1.40 − 0.319i)14-s + (−0.317 + 0.0725i)15-s + (−2.05 − 2.57i)16-s + 5.18i·17-s + ⋯
L(s)  = 1  + (0.897 + 0.204i)2-s + (0.250 + 0.520i)3-s + (−0.137 − 0.0660i)4-s + (−0.0324 + 0.142i)5-s + (0.118 + 0.518i)6-s + (0.375 − 0.180i)7-s + (−0.829 − 0.661i)8-s + (−0.207 + 0.260i)9-s + (−0.0582 + 0.120i)10-s + (−0.181 + 0.144i)11-s − 0.0878i·12-s + (−0.831 − 1.04i)13-s + (0.374 − 0.0854i)14-s + (−0.0820 + 0.0187i)15-s + (−0.514 − 0.644i)16-s + 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.893 - 0.449i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ 0.893 - 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34118 + 0.318159i\)
\(L(\frac12)\) \(\approx\) \(1.34118 + 0.318159i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.433 - 0.900i)T \)
29 \( 1 + (0.401 - 5.37i)T \)
good2 \( 1 + (-1.26 - 0.289i)T + (1.80 + 0.867i)T^{2} \)
5 \( 1 + (0.0725 - 0.317i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (-0.994 + 0.478i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (0.600 - 0.478i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (2.99 + 3.76i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 - 5.18iT - 17T^{2} \)
19 \( 1 + (-1.81 + 3.77i)T + (-11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.314 - 1.37i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (4.40 + 1.00i)T + (27.9 + 13.4i)T^{2} \)
37 \( 1 + (-1.40 - 1.11i)T + (8.23 + 36.0i)T^{2} \)
41 \( 1 - 1.34iT - 41T^{2} \)
43 \( 1 + (-7.02 + 1.60i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (-0.354 + 0.282i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.96 + 8.60i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 - 3.67T + 59T^{2} \)
61 \( 1 + (-3.83 - 7.96i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (-9.51 + 11.9i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (9.37 + 11.7i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (13.3 - 3.04i)T + (65.7 - 31.6i)T^{2} \)
79 \( 1 + (4.56 + 3.63i)T + (17.5 + 77.0i)T^{2} \)
83 \( 1 + (11.4 + 5.52i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (8.07 + 1.84i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + (0.670 - 1.39i)T + (-60.4 - 75.8i)T^{2} \)
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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

−14.57944817648183980448129438559, −13.25597097683668551746520432427, −12.54210321372947978581971640040, −11.00970266803790011406846211975, −9.957326847891608073153128730310, −8.758771086862116362586785625035, −7.29813027804113955516266212406, −5.62408149344739055288919525422, −4.64466726082141045566983764374, −3.21812271479302537499477625860, 2.59533023758488096287048152454, 4.33392523373970855116825821044, 5.56128335251870791597856822408, 7.17695588135637479042330621045, 8.492142968713464943450685348027, 9.569866285800779113002395579299, 11.43911593086508153930220580422, 12.15029478394121655943407277853, 13.06885392611641932492238087046, 14.22099605900451846558505939381

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