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Properties

Label 2-87-87.86-c4-0-14
Degree $2$
Conductor $87$
Sign $0.938 - 0.345i$
Analytic cond. $8.99318$
Root an. cond. $2.99886$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.36·2-s + (−1.79 − 8.81i)3-s − 14.1·4-s + 35.6i·5-s + (−2.44 − 12.0i)6-s + 75.8·7-s − 41.0·8-s + (−74.5 + 31.6i)9-s + 48.5i·10-s + 182.·11-s + (25.3 + 124. i)12-s − 13.7·13-s + 103.·14-s + (314. − 63.9i)15-s + 170.·16-s + 243.·17-s + ⋯
L(s)  = 1  + 0.340·2-s + (−0.199 − 0.979i)3-s − 0.884·4-s + 1.42i·5-s + (−0.0677 − 0.333i)6-s + 1.54·7-s − 0.641·8-s + (−0.920 + 0.390i)9-s + 0.485i·10-s + 1.51·11-s + (0.175 + 0.866i)12-s − 0.0814·13-s + 0.527·14-s + (1.39 − 0.284i)15-s + 0.665·16-s + 0.842·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.938 - 0.345i$
Analytic conductor: \(8.99318\)
Root analytic conductor: \(2.99886\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :2),\ 0.938 - 0.345i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.72101 + 0.306727i\)
\(L(\frac12)\) \(\approx\) \(1.72101 + 0.306727i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.79 + 8.81i)T \)
29 \( 1 + (-127. - 831. i)T \)
good2 \( 1 - 1.36T + 16T^{2} \)
5 \( 1 - 35.6iT - 625T^{2} \)
7 \( 1 - 75.8T + 2.40e3T^{2} \)
11 \( 1 - 182.T + 1.46e4T^{2} \)
13 \( 1 + 13.7T + 2.85e4T^{2} \)
17 \( 1 - 243.T + 8.35e4T^{2} \)
19 \( 1 - 193. iT - 1.30e5T^{2} \)
23 \( 1 - 143. iT - 2.79e5T^{2} \)
31 \( 1 - 1.00e3iT - 9.23e5T^{2} \)
37 \( 1 + 314. iT - 1.87e6T^{2} \)
41 \( 1 + 2.74e3T + 2.82e6T^{2} \)
43 \( 1 + 1.42e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.96e3T + 4.87e6T^{2} \)
53 \( 1 - 2.90e3iT - 7.89e6T^{2} \)
59 \( 1 + 4.80e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.90e3iT - 1.38e7T^{2} \)
67 \( 1 - 134.T + 2.01e7T^{2} \)
71 \( 1 - 6.13e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.85e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.12e4iT - 3.89e7T^{2} \)
83 \( 1 - 265. iT - 4.74e7T^{2} \)
89 \( 1 + 1.09e3T + 6.27e7T^{2} \)
97 \( 1 - 1.12e4iT - 8.85e7T^{2} \)
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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

−14.03203765407061137212744198933, −12.30752773561721496401538376076, −11.60716644842565876724946773431, −10.50199923116661428818492652411, −8.821540051274247898009769239297, −7.69375387399327663370680039505, −6.55559128774223917659023228089, −5.22172436745015585885498725399, −3.48420740555332770978454407161, −1.49525197269022294579243395823, 0.979038120421890247997467920502, 4.08360107187990780589573377510, 4.71903634276541686565956015679, 5.66574631962123300663206833456, 8.253807994677084073092431188921, 8.916680617501240774691651099983, 9.843878318503980040841263940656, 11.54922212606175730303961517434, 12.11924056823961793993435426459, 13.57899148418531303893383836359

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