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Properties

Label 2-87-87.86-c4-0-5
Degree $2$
Conductor $87$
Sign $-0.991 + 0.128i$
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  − 5.73·2-s + (7.98 + 4.14i)3-s + 16.9·4-s + 47.2i·5-s + (−45.8 − 23.7i)6-s − 77.2·7-s − 5.28·8-s + (46.6 + 66.2i)9-s − 270. i·10-s + 9.19·11-s + (135. + 70.1i)12-s + 132.·13-s + 443.·14-s + (−195. + 377. i)15-s − 240.·16-s − 152.·17-s + ⋯
L(s)  = 1  − 1.43·2-s + (0.887 + 0.460i)3-s + 1.05·4-s + 1.88i·5-s + (−1.27 − 0.660i)6-s − 1.57·7-s − 0.0825·8-s + (0.576 + 0.817i)9-s − 2.70i·10-s + 0.0760·11-s + (0.938 + 0.486i)12-s + 0.785·13-s + 2.26·14-s + (−0.869 + 1.67i)15-s − 0.939·16-s − 0.526·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.991 + 0.128i$
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: \((2,\ 87,\ (\ :2),\ -0.991 + 0.128i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0343755 - 0.532317i\)
\(L(\frac12)\) \(\approx\) \(0.0343755 - 0.532317i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-7.98 - 4.14i)T \)
29 \( 1 + (690. - 479. i)T \)
good2 \( 1 + 5.73T + 16T^{2} \)
5 \( 1 - 47.2iT - 625T^{2} \)
7 \( 1 + 77.2T + 2.40e3T^{2} \)
11 \( 1 - 9.19T + 1.46e4T^{2} \)
13 \( 1 - 132.T + 2.85e4T^{2} \)
17 \( 1 + 152.T + 8.35e4T^{2} \)
19 \( 1 + 482. iT - 1.30e5T^{2} \)
23 \( 1 + 175. iT - 2.79e5T^{2} \)
31 \( 1 - 848. iT - 9.23e5T^{2} \)
37 \( 1 - 177. iT - 1.87e6T^{2} \)
41 \( 1 + 1.78e3T + 2.82e6T^{2} \)
43 \( 1 + 1.47e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.26e3T + 4.87e6T^{2} \)
53 \( 1 - 3.70e3iT - 7.89e6T^{2} \)
59 \( 1 + 454. iT - 1.21e7T^{2} \)
61 \( 1 + 6.03e3iT - 1.38e7T^{2} \)
67 \( 1 - 1.24e3T + 2.01e7T^{2} \)
71 \( 1 - 5.50e3iT - 2.54e7T^{2} \)
73 \( 1 - 8.87e3iT - 2.83e7T^{2} \)
79 \( 1 - 956. iT - 3.89e7T^{2} \)
83 \( 1 - 1.64e3iT - 4.74e7T^{2} \)
89 \( 1 - 417.T + 6.27e7T^{2} \)
97 \( 1 - 266. iT - 8.85e7T^{2} \)
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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

−14.09243780387396712948341362847, −13.23807647869303385508146730956, −11.08310010881596999238592291881, −10.44650045184968345873200261397, −9.607730421326606663392036320536, −8.726362810913341618331622922559, −7.20853954720893649630174818790, −6.63272448132976042317360640704, −3.55559520447927831562079016183, −2.49831211832050767482362137660, 0.36729035971958122696852550442, 1.66929251359731825551531121019, 3.90658318814094185048284149050, 6.16849840829640255031869826418, 7.70622140558162102235643359872, 8.623990087012099854710948008224, 9.302244502594383330239169093417, 9.967633622709729890011188197677, 11.95177731142594304068248044639, 13.08189721229683915254667075530

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