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Properties

Label 2-87-29.23-c1-0-0
Degree $2$
Conductor $87$
Sign $-0.881 - 0.472i$
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 + 1.71i)2-s + (0.222 − 0.974i)3-s + (−0.628 − 2.75i)4-s + (−2.54 + 3.18i)5-s + (1.36 + 1.71i)6-s + (−0.811 + 3.55i)7-s + (1.63 + 0.786i)8-s + (−0.900 − 0.433i)9-s + (−1.99 − 8.73i)10-s + (3.17 − 1.53i)11-s − 2.82·12-s + (−0.834 + 0.402i)13-s + (−4.99 − 6.26i)14-s + (2.54 + 3.18i)15-s + (1.50 − 0.724i)16-s + 1.61·17-s + ⋯
L(s)  = 1  + (−0.968 + 1.21i)2-s + (0.128 − 0.562i)3-s + (−0.314 − 1.37i)4-s + (−1.13 + 1.42i)5-s + (0.559 + 0.701i)6-s + (−0.306 + 1.34i)7-s + (0.577 + 0.277i)8-s + (−0.300 − 0.144i)9-s + (−0.630 − 2.76i)10-s + (0.958 − 0.461i)11-s − 0.815·12-s + (−0.231 + 0.111i)13-s + (−1.33 − 1.67i)14-s + (0.656 + 0.823i)15-s + (0.376 − 0.181i)16-s + 0.392·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.881 - 0.472i$
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: \((2,\ 87,\ (\ :1/2),\ -0.881 - 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115121 + 0.458901i\)
\(L(\frac12)\) \(\approx\) \(0.115121 + 0.458901i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.19 + 1.43i)T \)
good2 \( 1 + (1.36 - 1.71i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (2.54 - 3.18i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.811 - 3.55i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (-3.17 + 1.53i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.834 - 0.402i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 - 1.61T + 17T^{2} \)
19 \( 1 + (-0.886 - 3.88i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (-0.963 - 1.20i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (1.00 - 1.25i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (-4.77 - 2.30i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + 6.71T + 41T^{2} \)
43 \( 1 + (-2.76 - 3.46i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (3.00 - 1.44i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-1.27 + 1.59i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 - 7.29T + 59T^{2} \)
61 \( 1 + (-1.33 + 5.85i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (8.54 + 4.11i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (7.61 - 3.66i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-4.71 - 5.91i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (0.259 + 0.124i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (-1.44 - 6.31i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-10.2 + 12.9i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (2.23 + 9.80i)T + (-87.3 + 42.0i)T^{2} \)
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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

−14.84494962934818121891519417654, −14.27757045370274437708462084493, −12.22238145556166435774378668696, −11.52106907446574385740040894749, −9.931916942921962243781696739186, −8.672680076593385096086781910154, −7.82421778983140250247807195449, −6.76954759821916524734056361812, −6.01415881187197505451899146134, −3.18949509739808649516602472934, 0.840602669496891280351210769258, 3.62433848994998155292623325916, 4.59451425935125874738159811682, 7.39727395962489050433871042774, 8.580606293894004414992790591931, 9.400053005328048438913470314270, 10.39695854605230818572181306643, 11.52254472677245687643893826720, 12.26080821337876697931680632070, 13.33328786289006017623822044689

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