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Properties

Label 2-87-87.41-c1-0-6
Degree $2$
Conductor $87$
Sign $0.755 + 0.654i$
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  + (1.36 − 1.36i)2-s + (1.5 + 0.866i)3-s − 1.73i·4-s − 3.73·5-s + (3.23 − 0.866i)6-s − 2.73·7-s + (0.366 + 0.366i)8-s + (1.5 + 2.59i)9-s + (−5.09 + 5.09i)10-s + (0.366 − 0.366i)11-s + (1.49 − 2.59i)12-s − 5.73i·13-s + (−3.73 + 3.73i)14-s + (−5.59 − 3.23i)15-s + 4.46·16-s + (0.732 − 0.732i)17-s + ⋯
L(s)  = 1  + (0.965 − 0.965i)2-s + (0.866 + 0.499i)3-s − 0.866i·4-s − 1.66·5-s + (1.31 − 0.353i)6-s − 1.03·7-s + (0.129 + 0.129i)8-s + (0.5 + 0.866i)9-s + (−1.61 + 1.61i)10-s + (0.110 − 0.110i)11-s + (0.433 − 0.749i)12-s − 1.58i·13-s + (−0.997 + 0.997i)14-s + (−1.44 − 0.834i)15-s + 1.11·16-s + (0.177 − 0.177i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.755 + 0.654i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ 0.755 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39983 - 0.521996i\)
\(L(\frac12)\) \(\approx\) \(1.39983 - 0.521996i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
29 \( 1 + (2 - 5i)T \)
good2 \( 1 + (-1.36 + 1.36i)T - 2iT^{2} \)
5 \( 1 + 3.73T + 5T^{2} \)
7 \( 1 + 2.73T + 7T^{2} \)
11 \( 1 + (-0.366 + 0.366i)T - 11iT^{2} \)
13 \( 1 + 5.73iT - 13T^{2} \)
17 \( 1 + (-0.732 + 0.732i)T - 17iT^{2} \)
19 \( 1 + (-1.73 - 1.73i)T + 19iT^{2} \)
23 \( 1 - 1.46iT - 23T^{2} \)
31 \( 1 + (-0.633 - 0.633i)T + 31iT^{2} \)
37 \( 1 + (3.46 - 3.46i)T - 37iT^{2} \)
41 \( 1 + (5.19 + 5.19i)T + 41iT^{2} \)
43 \( 1 + (3.56 + 3.56i)T + 43iT^{2} \)
47 \( 1 + (0.830 + 0.830i)T + 47iT^{2} \)
53 \( 1 - 3.92iT - 53T^{2} \)
59 \( 1 + 0.535iT - 59T^{2} \)
61 \( 1 + (2 + 2i)T + 61iT^{2} \)
67 \( 1 + 8.92iT - 67T^{2} \)
71 \( 1 - 16.7T + 71T^{2} \)
73 \( 1 + (4 - 4i)T - 73iT^{2} \)
79 \( 1 + (-1.83 - 1.83i)T + 79iT^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (5 - 5i)T - 89iT^{2} \)
97 \( 1 + (7.73 - 7.73i)T - 97iT^{2} \)
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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

−13.90282884364911597863585403952, −12.85045237698661652512310440888, −12.18333956701487897777465826254, −10.96355091930756574913806723462, −10.05019879989224493084680105528, −8.431649014375362330618434841101, −7.45697501640839057994477981531, −5.07798814371478233673322338487, −3.56578793271638368759191144750, −3.24335062228152316731830523739, 3.47282394528377000084870382426, 4.36455706855295643630641564492, 6.54706368023376339910757860235, 7.18644921415016326001496936084, 8.265616017608877052600928627605, 9.609116065850912031873943077382, 11.62853933355564427212395679818, 12.52575051499861597875557768621, 13.45442827559578591021338930418, 14.42970647977292479545655439178

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