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Properties

Label 2-87-29.23-c1-0-4
Degree $2$
Conductor $87$
Sign $0.290 + 0.956i$
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  + (1.14 − 1.43i)2-s + (0.222 − 0.974i)3-s + (−0.303 − 1.33i)4-s + (−1.40 + 1.76i)5-s + (−1.14 − 1.43i)6-s + (0.165 − 0.725i)7-s + (1.04 + 0.504i)8-s + (−0.900 − 0.433i)9-s + (0.922 + 4.03i)10-s + (−5.68 + 2.73i)11-s − 1.36·12-s + (1.48 − 0.714i)13-s + (−0.851 − 1.06i)14-s + (1.40 + 1.76i)15-s + (4.38 − 2.11i)16-s + 4.16·17-s + ⋯
L(s)  = 1  + (0.808 − 1.01i)2-s + (0.128 − 0.562i)3-s + (−0.151 − 0.665i)4-s + (−0.629 + 0.789i)5-s + (−0.467 − 0.585i)6-s + (0.0625 − 0.274i)7-s + (0.370 + 0.178i)8-s + (−0.300 − 0.144i)9-s + (0.291 + 1.27i)10-s + (−1.71 + 0.825i)11-s − 0.394·12-s + (0.411 − 0.198i)13-s + (−0.227 − 0.285i)14-s + (0.363 + 0.455i)15-s + (1.09 − 0.527i)16-s + 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.290 + 0.956i$
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: \((2,\ 87,\ (\ :1/2),\ 0.290 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07338 - 0.795575i\)
\(L(\frac12)\) \(\approx\) \(1.07338 - 0.795575i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.68 + 2.65i)T \)
good2 \( 1 + (-1.14 + 1.43i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (1.40 - 1.76i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (-0.165 + 0.725i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (5.68 - 2.73i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-1.48 + 0.714i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 - 4.16T + 17T^{2} \)
19 \( 1 + (0.231 + 1.01i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (2.85 + 3.57i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (2.20 - 2.76i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (-0.969 - 0.466i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + 1.20T + 41T^{2} \)
43 \( 1 + (7.99 + 10.0i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-7.10 + 3.42i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (8.22 - 10.3i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 + (-1.43 + 6.28i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (-6.19 - 2.98i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-2.95 + 1.42i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-4.42 - 5.54i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (5.75 + 2.77i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (-1.14 - 5.02i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-5.80 + 7.28i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-3.37 - 14.7i)T + (-87.3 + 42.0i)T^{2} \)
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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

−13.68477986512241917689039189821, −12.81938882166421447036056916606, −12.02050504283647730236679127649, −10.89013865084419211115554209392, −10.22060011920424518086121041236, −8.008697569370666197617668340366, −7.25640684521460144674291903025, −5.31148973087099081828135633612, −3.68821588661257684820173630756, −2.42521494632738723457307893262, 3.64903650112333742707224865203, 5.05181233627449719491525356057, 5.79281001245314802768439148892, 7.70229613149571380523816029499, 8.388187763864793944217306503595, 10.01382899820837782735571331021, 11.27699900407404653424500684347, 12.68595656100089856444479119597, 13.50330128522437668413479190721, 14.59190111820962048798259125435

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