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Properties

Label 2-87-29.24-c1-0-5
Degree 22
Conductor 8787
Sign 0.474+0.880i0.474 + 0.880i
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  + (0.110 + 0.138i)2-s + (−0.222 − 0.974i)3-s + (0.438 − 1.91i)4-s + (−2.15 − 2.69i)5-s + (0.110 − 0.138i)6-s + (0.844 + 3.70i)7-s + (0.633 − 0.304i)8-s + (−0.900 + 0.433i)9-s + (0.135 − 0.595i)10-s + (3.84 + 1.85i)11-s − 1.96·12-s + (4.18 + 2.01i)13-s + (−0.419 + 0.525i)14-s + (−2.15 + 2.69i)15-s + (−3.43 − 1.65i)16-s − 3.07·17-s + ⋯
L(s)  = 1  + (0.0780 + 0.0978i)2-s + (−0.128 − 0.562i)3-s + (0.219 − 0.959i)4-s + (−0.962 − 1.20i)5-s + (0.0450 − 0.0565i)6-s + (0.319 + 1.39i)7-s + (0.223 − 0.107i)8-s + (−0.300 + 0.144i)9-s + (0.0430 − 0.188i)10-s + (1.15 + 0.558i)11-s − 0.568·12-s + (1.16 + 0.558i)13-s + (−0.111 + 0.140i)14-s + (−0.555 + 0.696i)15-s + (−0.858 − 0.413i)16-s − 0.745·17-s + ⋯

Functional equation

Λ(s)=(87s/2 ΓC(s) L(s)=((0.474+0.880i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(87s/2 ΓC(s+1/2) L(s)=((0.474+0.880i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8787    =    3⋅293 \cdot 29
Sign: 0.474+0.880i0.474 + 0.880i
Analytic conductor: 0.6946980.694698
Root analytic conductor: 0.8334850.833485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ87(82,⋅)\chi_{87} (82, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 87, ( :1/2), 0.474+0.880i)(2,\ 87,\ (\ :1/2),\ 0.474 + 0.880i)

Particular Values

L(1)L(1) ≈\approx 0.813378−0.485387i0.813378 - 0.485387i
L(12)L(\frac12) ≈\approx 0.813378−0.485387i0.813378 - 0.485387i
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+(1.41−5.19i)T 1 + (1.41 - 5.19i)T
good2 1+(−0.110−0.138i)T+(−0.445+1.94i)T2 1 + (-0.110 - 0.138i)T + (-0.445 + 1.94i)T^{2}
5 1+(2.15+2.69i)T+(−1.11+4.87i)T2 1 + (2.15 + 2.69i)T + (-1.11 + 4.87i)T^{2}
7 1+(−0.844−3.70i)T+(−6.30+3.03i)T2 1 + (-0.844 - 3.70i)T + (-6.30 + 3.03i)T^{2}
11 1+(−3.84−1.85i)T+(6.85+8.60i)T2 1 + (-3.84 - 1.85i)T + (6.85 + 8.60i)T^{2}
13 1+(−4.18−2.01i)T+(8.10+10.1i)T2 1 + (-4.18 - 2.01i)T + (8.10 + 10.1i)T^{2}
17 1+3.07T+17T2 1 + 3.07T + 17T^{2}
19 1+(−0.799+3.50i)T+(−17.1−8.24i)T2 1 + (-0.799 + 3.50i)T + (-17.1 - 8.24i)T^{2}
23 1+(0.270−0.339i)T+(−5.11−22.4i)T2 1 + (0.270 - 0.339i)T + (-5.11 - 22.4i)T^{2}
31 1+(2.81+3.53i)T+(−6.89+30.2i)T2 1 + (2.81 + 3.53i)T + (-6.89 + 30.2i)T^{2}
37 1+(5.70−2.74i)T+(23.0−28.9i)T2 1 + (5.70 - 2.74i)T + (23.0 - 28.9i)T^{2}
41 1+1.97T+41T2 1 + 1.97T + 41T^{2}
43 1+(0.156−0.196i)T+(−9.56−41.9i)T2 1 + (0.156 - 0.196i)T + (-9.56 - 41.9i)T^{2}
47 1+(−4.33−2.08i)T+(29.3+36.7i)T2 1 + (-4.33 - 2.08i)T + (29.3 + 36.7i)T^{2}
53 1+(−6.83−8.57i)T+(−11.7+51.6i)T2 1 + (-6.83 - 8.57i)T + (-11.7 + 51.6i)T^{2}
59 1+6.06T+59T2 1 + 6.06T + 59T^{2}
61 1+(0.843+3.69i)T+(−54.9+26.4i)T2 1 + (0.843 + 3.69i)T + (-54.9 + 26.4i)T^{2}
67 1+(−4.74+2.28i)T+(41.7−52.3i)T2 1 + (-4.74 + 2.28i)T + (41.7 - 52.3i)T^{2}
71 1+(−5.25−2.53i)T+(44.2+55.5i)T2 1 + (-5.25 - 2.53i)T + (44.2 + 55.5i)T^{2}
73 1+(6.57−8.24i)T+(−16.2−71.1i)T2 1 + (6.57 - 8.24i)T + (-16.2 - 71.1i)T^{2}
79 1+(9.04−4.35i)T+(49.2−61.7i)T2 1 + (9.04 - 4.35i)T + (49.2 - 61.7i)T^{2}
83 1+(−1.57+6.92i)T+(−74.7−36.0i)T2 1 + (-1.57 + 6.92i)T + (-74.7 - 36.0i)T^{2}
89 1+(8.71+10.9i)T+(−19.8+86.7i)T2 1 + (8.71 + 10.9i)T + (-19.8 + 86.7i)T^{2}
97 1+(−3.18+13.9i)T+(−87.3−42.0i)T2 1 + (-3.18 + 13.9i)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

−14.05837496491112017887779645986, −12.78559436684390851059278183530, −11.80338145237446541604635134830, −11.25771287304732653181834233074, −9.115843622506061739692580368098, −8.704512754058978002650269533933, −6.95744927152153872264786266546, −5.70557473956735926140824629900, −4.48230233580441468873033632859, −1.59293336240047390911440003872, 3.60024090728967690911123052478, 3.91975088207769563914483179072, 6.53872968387328789410731196416, 7.51929970288157838660452618717, 8.587450755224876100517384035861, 10.51994037809613692176538014627, 11.10553710782365320729925803066, 11.86858572639326942896405778249, 13.49423844387374654129049164042, 14.34523316707992453157793477283

Graph of the ZZ-function along the critical line