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Properties

Label 2-87-29.28-c1-0-0
Degree $2$
Conductor $87$
Sign $0.557 - 0.830i$
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  + 0.618i·2-s + i·3-s + 1.61·4-s − 1.23·5-s − 0.618·6-s + 0.236·7-s + 2.23i·8-s − 9-s − 0.763i·10-s − 2.23i·11-s + 1.61i·12-s − 13-s + 0.145i·14-s − 1.23i·15-s + 1.85·16-s − 3i·17-s + ⋯
L(s)  = 1  + 0.437i·2-s + 0.577i·3-s + 0.809·4-s − 0.552·5-s − 0.252·6-s + 0.0892·7-s + 0.790i·8-s − 0.333·9-s − 0.241i·10-s − 0.674i·11-s + 0.467i·12-s − 0.277·13-s + 0.0389i·14-s − 0.319i·15-s + 0.463·16-s − 0.727i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.557 - 0.830i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ 0.557 - 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.924735 + 0.493197i\)
\(L(\frac12)\) \(\approx\) \(0.924735 + 0.493197i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
29 \( 1 + (-4.47 - 3i)T \)
good2 \( 1 - 0.618iT - 2T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 - 0.236T + 7T^{2} \)
11 \( 1 + 2.23iT - 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + 5.70iT - 19T^{2} \)
23 \( 1 + 3.23T + 23T^{2} \)
31 \( 1 + 2.76iT - 31T^{2} \)
37 \( 1 - 9.23iT - 37T^{2} \)
41 \( 1 - 4.47iT - 41T^{2} \)
43 \( 1 - 8.47iT - 43T^{2} \)
47 \( 1 + 5.76iT - 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 - 7.23iT - 61T^{2} \)
67 \( 1 + 8.70T + 67T^{2} \)
71 \( 1 - 9.23T + 71T^{2} \)
73 \( 1 - 5.70iT - 73T^{2} \)
79 \( 1 + 15.7iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 17.9iT - 89T^{2} \)
97 \( 1 + 4.18iT - 97T^{2} \)
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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

−14.67096616241862822894929310969, −13.52028951706159641456029378348, −11.81541397245759299695200739327, −11.32242339419112691341998908021, −10.07391762188405516333126232330, −8.616094809307978180108285806494, −7.52319752337265024985672880202, −6.25210055772379311908766450576, −4.80171995312976610053413441723, −2.97038936806824430773598937284, 2.00718633194260351255256283593, 3.84268687105472536740287733842, 5.96468628375135172750882048602, 7.23660794278294829129180385198, 8.132810516947386341120299804898, 9.908678994501687779211182638847, 10.92493536013980766844073253122, 12.21812210801941119359865164619, 12.37674484358416324759885608696, 14.00309307262373821971562443762

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