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Properties

Label 2-87-29.23-c1-0-4
Degree 22
Conductor 8787
Sign 0.290+0.956i0.290 + 0.956i
Analytic cond. 0.6946980.694698
Root an. cond. 0.8334850.833485
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(87s/2 ΓC(s) L(s)=((0.290+0.956i) Λ‾(2−s)\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}
Λ(s)=(87s/2 ΓC(s+1/2) L(s)=((0.290+0.956i) Λ‾(1−s)\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: 22
Conductor: 8787    =    3⋅293 \cdot 29
Sign: 0.290+0.956i0.290 + 0.956i
Analytic conductor: 0.6946980.694698
Root analytic conductor: 0.8334850.833485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ87(52,⋅)\chi_{87} (52, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 87, ( :1/2), 0.290+0.956i)(2,\ 87,\ (\ :1/2),\ 0.290 + 0.956i)

Particular Values

L(1)L(1) ≈\approx 1.07338−0.795575i1.07338 - 0.795575i
L(12)L(\frac12) ≈\approx 1.07338−0.795575i1.07338 - 0.795575i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(−0.222+0.974i)T 1 + (-0.222 + 0.974i)T
29 1+(4.68+2.65i)T 1 + (4.68 + 2.65i)T
good2 1+(−1.14+1.43i)T+(−0.445−1.94i)T2 1 + (-1.14 + 1.43i)T + (-0.445 - 1.94i)T^{2}
5 1+(1.40−1.76i)T+(−1.11−4.87i)T2 1 + (1.40 - 1.76i)T + (-1.11 - 4.87i)T^{2}
7 1+(−0.165+0.725i)T+(−6.30−3.03i)T2 1 + (-0.165 + 0.725i)T + (-6.30 - 3.03i)T^{2}
11 1+(5.68−2.73i)T+(6.85−8.60i)T2 1 + (5.68 - 2.73i)T + (6.85 - 8.60i)T^{2}
13 1+(−1.48+0.714i)T+(8.10−10.1i)T2 1 + (-1.48 + 0.714i)T + (8.10 - 10.1i)T^{2}
17 1−4.16T+17T2 1 - 4.16T + 17T^{2}
19 1+(0.231+1.01i)T+(−17.1+8.24i)T2 1 + (0.231 + 1.01i)T + (-17.1 + 8.24i)T^{2}
23 1+(2.85+3.57i)T+(−5.11+22.4i)T2 1 + (2.85 + 3.57i)T + (-5.11 + 22.4i)T^{2}
31 1+(2.20−2.76i)T+(−6.89−30.2i)T2 1 + (2.20 - 2.76i)T + (-6.89 - 30.2i)T^{2}
37 1+(−0.969−0.466i)T+(23.0+28.9i)T2 1 + (-0.969 - 0.466i)T + (23.0 + 28.9i)T^{2}
41 1+1.20T+41T2 1 + 1.20T + 41T^{2}
43 1+(7.99+10.0i)T+(−9.56+41.9i)T2 1 + (7.99 + 10.0i)T + (-9.56 + 41.9i)T^{2}
47 1+(−7.10+3.42i)T+(29.3−36.7i)T2 1 + (-7.10 + 3.42i)T + (29.3 - 36.7i)T^{2}
53 1+(8.22−10.3i)T+(−11.7−51.6i)T2 1 + (8.22 - 10.3i)T + (-11.7 - 51.6i)T^{2}
59 1−11.0T+59T2 1 - 11.0T + 59T^{2}
61 1+(−1.43+6.28i)T+(−54.9−26.4i)T2 1 + (-1.43 + 6.28i)T + (-54.9 - 26.4i)T^{2}
67 1+(−6.19−2.98i)T+(41.7+52.3i)T2 1 + (-6.19 - 2.98i)T + (41.7 + 52.3i)T^{2}
71 1+(−2.95+1.42i)T+(44.2−55.5i)T2 1 + (-2.95 + 1.42i)T + (44.2 - 55.5i)T^{2}
73 1+(−4.42−5.54i)T+(−16.2+71.1i)T2 1 + (-4.42 - 5.54i)T + (-16.2 + 71.1i)T^{2}
79 1+(5.75+2.77i)T+(49.2+61.7i)T2 1 + (5.75 + 2.77i)T + (49.2 + 61.7i)T^{2}
83 1+(−1.14−5.02i)T+(−74.7+36.0i)T2 1 + (-1.14 - 5.02i)T + (-74.7 + 36.0i)T^{2}
89 1+(−5.80+7.28i)T+(−19.8−86.7i)T2 1 + (-5.80 + 7.28i)T + (-19.8 - 86.7i)T^{2}
97 1+(−3.37−14.7i)T+(−87.3+42.0i)T2 1 + (-3.37 - 14.7i)T + (-87.3 + 42.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line