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Properties

Label 2-87-29.23-c1-0-2
Degree $2$
Conductor $87$
Sign $0.474 - 0.880i$
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  + (0.110 − 0.138i)2-s + (−0.222 + 0.974i)3-s + (0.438 + 1.91i)4-s + (−2.15 + 2.69i)5-s + (0.110 + 0.138i)6-s + (0.844 − 3.70i)7-s + (0.633 + 0.304i)8-s + (−0.900 − 0.433i)9-s + (0.135 + 0.595i)10-s + (3.84 − 1.85i)11-s − 1.96·12-s + (4.18 − 2.01i)13-s + (−0.419 − 0.525i)14-s + (−2.15 − 2.69i)15-s + (−3.43 + 1.65i)16-s − 3.07·17-s + ⋯
L(s)  = 1  + (0.0780 − 0.0978i)2-s + (−0.128 + 0.562i)3-s + (0.219 + 0.959i)4-s + (−0.962 + 1.20i)5-s + (0.0450 + 0.0565i)6-s + (0.319 − 1.39i)7-s + (0.223 + 0.107i)8-s + (−0.300 − 0.144i)9-s + (0.0430 + 0.188i)10-s + (1.15 − 0.558i)11-s − 0.568·12-s + (1.16 − 0.558i)13-s + (−0.111 − 0.140i)14-s + (−0.555 − 0.696i)15-s + (−0.858 + 0.413i)16-s − 0.745·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.474 - 0.880i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ 0.474 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.813378 + 0.485387i\)
\(L(\frac12)\) \(\approx\) \(0.813378 + 0.485387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.222 - 0.974i)T \)
29 \( 1 + (1.41 + 5.19i)T \)
good2 \( 1 + (-0.110 + 0.138i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (2.15 - 2.69i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (-0.844 + 3.70i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (-3.84 + 1.85i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-4.18 + 2.01i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + 3.07T + 17T^{2} \)
19 \( 1 + (-0.799 - 3.50i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (0.270 + 0.339i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (2.81 - 3.53i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (5.70 + 2.74i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + 1.97T + 41T^{2} \)
43 \( 1 + (0.156 + 0.196i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-4.33 + 2.08i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-6.83 + 8.57i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + 6.06T + 59T^{2} \)
61 \( 1 + (0.843 - 3.69i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (-4.74 - 2.28i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-5.25 + 2.53i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (6.57 + 8.24i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (9.04 + 4.35i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (-1.57 - 6.92i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (8.71 - 10.9i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-3.18 - 13.9i)T + (-87.3 + 42.0i)T^{2} \)
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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

−14.34523316707992453157793477283, −13.49423844387374654129049164042, −11.86858572639326942896405778249, −11.10553710782365320729925803066, −10.51994037809613692176538014627, −8.587450755224876100517384035861, −7.51929970288157838660452618717, −6.53872968387328789410731196416, −3.91975088207769563914483179072, −3.60024090728967690911123052478, 1.59293336240047390911440003872, 4.48230233580441468873033632859, 5.70557473956735926140824629900, 6.95744927152153872264786266546, 8.704512754058978002650269533933, 9.115843622506061739692580368098, 11.25771287304732653181834233074, 11.80338145237446541604635134830, 12.78559436684390851059278183530, 14.05837496491112017887779645986

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