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Properties

Label 2-87-29.7-c1-0-3
Degree 22
Conductor 8787
Sign 0.894+0.446i0.894 + 0.446i
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  + (0.0321 + 0.140i)2-s + (−0.900 − 0.433i)3-s + (1.78 − 0.858i)4-s + (−0.345 − 1.51i)5-s + (0.0321 − 0.140i)6-s + (1.37 + 0.662i)7-s + (0.358 + 0.449i)8-s + (0.623 + 0.781i)9-s + (0.202 − 0.0973i)10-s + (−0.478 + 0.600i)11-s − 1.97·12-s + (−1.63 + 2.04i)13-s + (−0.0490 + 0.215i)14-s + (−0.345 + 1.51i)15-s + (2.41 − 3.02i)16-s − 3.51·17-s + ⋯
L(s)  = 1  + (0.0227 + 0.0995i)2-s + (−0.520 − 0.250i)3-s + (0.891 − 0.429i)4-s + (−0.154 − 0.677i)5-s + (0.0131 − 0.0574i)6-s + (0.520 + 0.250i)7-s + (0.126 + 0.158i)8-s + (0.207 + 0.260i)9-s + (0.0639 − 0.0307i)10-s + (−0.144 + 0.181i)11-s − 0.571·12-s + (−0.452 + 0.567i)13-s + (−0.0131 + 0.0574i)14-s + (−0.0892 + 0.391i)15-s + (0.604 − 0.757i)16-s − 0.852·17-s + ⋯

Functional equation

Λ(s)=(87s/2 ΓC(s) L(s)=((0.894+0.446i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(87s/2 ΓC(s+1/2) L(s)=((0.894+0.446i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8787    =    3⋅293 \cdot 29
Sign: 0.894+0.446i0.894 + 0.446i
Analytic conductor: 0.6946980.694698
Root analytic conductor: 0.8334850.833485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ87(7,⋅)\chi_{87} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 87, ( :1/2), 0.894+0.446i)(2,\ 87,\ (\ :1/2),\ 0.894 + 0.446i)

Particular Values

L(1)L(1) ≈\approx 0.966207−0.227907i0.966207 - 0.227907i
L(12)L(\frac12) ≈\approx 0.966207−0.227907i0.966207 - 0.227907i
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+(0.900+0.433i)T 1 + (0.900 + 0.433i)T
29 1+(−3.40+4.17i)T 1 + (-3.40 + 4.17i)T
good2 1+(−0.0321−0.140i)T+(−1.80+0.867i)T2 1 + (-0.0321 - 0.140i)T + (-1.80 + 0.867i)T^{2}
5 1+(0.345+1.51i)T+(−4.50+2.16i)T2 1 + (0.345 + 1.51i)T + (-4.50 + 2.16i)T^{2}
7 1+(−1.37−0.662i)T+(4.36+5.47i)T2 1 + (-1.37 - 0.662i)T + (4.36 + 5.47i)T^{2}
11 1+(0.478−0.600i)T+(−2.44−10.7i)T2 1 + (0.478 - 0.600i)T + (-2.44 - 10.7i)T^{2}
13 1+(1.63−2.04i)T+(−2.89−12.6i)T2 1 + (1.63 - 2.04i)T + (-2.89 - 12.6i)T^{2}
17 1+3.51T+17T2 1 + 3.51T + 17T^{2}
19 1+(4.72−2.27i)T+(11.8−14.8i)T2 1 + (4.72 - 2.27i)T + (11.8 - 14.8i)T^{2}
23 1+(1.89−8.28i)T+(−20.7−9.97i)T2 1 + (1.89 - 8.28i)T + (-20.7 - 9.97i)T^{2}
31 1+(−0.315−1.38i)T+(−27.9+13.4i)T2 1 + (-0.315 - 1.38i)T + (-27.9 + 13.4i)T^{2}
37 1+(1.87+2.35i)T+(−8.23+36.0i)T2 1 + (1.87 + 2.35i)T + (-8.23 + 36.0i)T^{2}
41 1−5.79T+41T2 1 - 5.79T + 41T^{2}
43 1+(0.955−4.18i)T+(−38.7−18.6i)T2 1 + (0.955 - 4.18i)T + (-38.7 - 18.6i)T^{2}
47 1+(−1.10+1.38i)T+(−10.4−45.8i)T2 1 + (-1.10 + 1.38i)T + (-10.4 - 45.8i)T^{2}
53 1+(1.50+6.60i)T+(−47.7+22.9i)T2 1 + (1.50 + 6.60i)T + (-47.7 + 22.9i)T^{2}
59 1−14.9T+59T2 1 - 14.9T + 59T^{2}
61 1+(12.3+5.95i)T+(38.0+47.6i)T2 1 + (12.3 + 5.95i)T + (38.0 + 47.6i)T^{2}
67 1+(4.54+5.70i)T+(−14.9+65.3i)T2 1 + (4.54 + 5.70i)T + (-14.9 + 65.3i)T^{2}
71 1+(2.99−3.76i)T+(−15.7−69.2i)T2 1 + (2.99 - 3.76i)T + (-15.7 - 69.2i)T^{2}
73 1+(2.04−8.96i)T+(−65.7−31.6i)T2 1 + (2.04 - 8.96i)T + (-65.7 - 31.6i)T^{2}
79 1+(−0.340−0.426i)T+(−17.5+77.0i)T2 1 + (-0.340 - 0.426i)T + (-17.5 + 77.0i)T^{2}
83 1+(10.4−5.05i)T+(51.7−64.8i)T2 1 + (10.4 - 5.05i)T + (51.7 - 64.8i)T^{2}
89 1+(3.06+13.4i)T+(−80.1+38.6i)T2 1 + (3.06 + 13.4i)T + (-80.1 + 38.6i)T^{2}
97 1+(−1.60+0.770i)T+(60.4−75.8i)T2 1 + (-1.60 + 0.770i)T + (60.4 - 75.8i)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

−14.22963467935774868816824210424, −12.84320591958800579238552009190, −11.85508620145153438733724530841, −11.13295682263897489253077756894, −9.874168875943196710144665590460, −8.356993459956776412929883665482, −7.10558541090672038179450544680, −5.89706204145986040926853953899, −4.64887843044907221215178461780, −1.92334256256028243820663637251, 2.67901717903275616668530623241, 4.47709277902001567762267936328, 6.30110974956992593324082866786, 7.22975190344445606786796815804, 8.544864067579210678659287418422, 10.62252392553050094239588145976, 10.75911588678619403186727746007, 12.00113659042793307969331366146, 12.95989344611590501622438245324, 14.55375078715726297853212233109

Graph of the ZZ-function along the critical line