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Properties

Label 2-87-29.24-c1-0-4
Degree 22
Conductor 8787
Sign −0.881+0.472i-0.881 + 0.472i
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  + (−1.36 − 1.71i)2-s + (0.222 + 0.974i)3-s + (−0.628 + 2.75i)4-s + (−2.54 − 3.18i)5-s + (1.36 − 1.71i)6-s + (−0.811 − 3.55i)7-s + (1.63 − 0.786i)8-s + (−0.900 + 0.433i)9-s + (−1.99 + 8.73i)10-s + (3.17 + 1.53i)11-s − 2.82·12-s + (−0.834 − 0.402i)13-s + (−4.99 + 6.26i)14-s + (2.54 − 3.18i)15-s + (1.50 + 0.724i)16-s + 1.61·17-s + ⋯
L(s)  = 1  + (−0.968 − 1.21i)2-s + (0.128 + 0.562i)3-s + (−0.314 + 1.37i)4-s + (−1.13 − 1.42i)5-s + (0.559 − 0.701i)6-s + (−0.306 − 1.34i)7-s + (0.577 − 0.277i)8-s + (−0.300 + 0.144i)9-s + (−0.630 + 2.76i)10-s + (0.958 + 0.461i)11-s − 0.815·12-s + (−0.231 − 0.111i)13-s + (−1.33 + 1.67i)14-s + (0.656 − 0.823i)15-s + (0.376 + 0.181i)16-s + 0.392·17-s + ⋯

Functional equation

Λ(s)=(87s/2 ΓC(s) L(s)=((−0.881+0.472i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(87s/2 ΓC(s+1/2) L(s)=((−0.881+0.472i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8787    =    3⋅293 \cdot 29
Sign: −0.881+0.472i-0.881 + 0.472i
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), −0.881+0.472i)(2,\ 87,\ (\ :1/2),\ -0.881 + 0.472i)

Particular Values

L(1)L(1) ≈\approx 0.115121−0.458901i0.115121 - 0.458901i
L(12)L(\frac12) ≈\approx 0.115121−0.458901i0.115121 - 0.458901i
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+(−5.19−1.43i)T 1 + (-5.19 - 1.43i)T
good2 1+(1.36+1.71i)T+(−0.445+1.94i)T2 1 + (1.36 + 1.71i)T + (-0.445 + 1.94i)T^{2}
5 1+(2.54+3.18i)T+(−1.11+4.87i)T2 1 + (2.54 + 3.18i)T + (-1.11 + 4.87i)T^{2}
7 1+(0.811+3.55i)T+(−6.30+3.03i)T2 1 + (0.811 + 3.55i)T + (-6.30 + 3.03i)T^{2}
11 1+(−3.17−1.53i)T+(6.85+8.60i)T2 1 + (-3.17 - 1.53i)T + (6.85 + 8.60i)T^{2}
13 1+(0.834+0.402i)T+(8.10+10.1i)T2 1 + (0.834 + 0.402i)T + (8.10 + 10.1i)T^{2}
17 1−1.61T+17T2 1 - 1.61T + 17T^{2}
19 1+(−0.886+3.88i)T+(−17.1−8.24i)T2 1 + (-0.886 + 3.88i)T + (-17.1 - 8.24i)T^{2}
23 1+(−0.963+1.20i)T+(−5.11−22.4i)T2 1 + (-0.963 + 1.20i)T + (-5.11 - 22.4i)T^{2}
31 1+(1.00+1.25i)T+(−6.89+30.2i)T2 1 + (1.00 + 1.25i)T + (-6.89 + 30.2i)T^{2}
37 1+(−4.77+2.30i)T+(23.0−28.9i)T2 1 + (-4.77 + 2.30i)T + (23.0 - 28.9i)T^{2}
41 1+6.71T+41T2 1 + 6.71T + 41T^{2}
43 1+(−2.76+3.46i)T+(−9.56−41.9i)T2 1 + (-2.76 + 3.46i)T + (-9.56 - 41.9i)T^{2}
47 1+(3.00+1.44i)T+(29.3+36.7i)T2 1 + (3.00 + 1.44i)T + (29.3 + 36.7i)T^{2}
53 1+(−1.27−1.59i)T+(−11.7+51.6i)T2 1 + (-1.27 - 1.59i)T + (-11.7 + 51.6i)T^{2}
59 1−7.29T+59T2 1 - 7.29T + 59T^{2}
61 1+(−1.33−5.85i)T+(−54.9+26.4i)T2 1 + (-1.33 - 5.85i)T + (-54.9 + 26.4i)T^{2}
67 1+(8.54−4.11i)T+(41.7−52.3i)T2 1 + (8.54 - 4.11i)T + (41.7 - 52.3i)T^{2}
71 1+(7.61+3.66i)T+(44.2+55.5i)T2 1 + (7.61 + 3.66i)T + (44.2 + 55.5i)T^{2}
73 1+(−4.71+5.91i)T+(−16.2−71.1i)T2 1 + (-4.71 + 5.91i)T + (-16.2 - 71.1i)T^{2}
79 1+(0.259−0.124i)T+(49.2−61.7i)T2 1 + (0.259 - 0.124i)T + (49.2 - 61.7i)T^{2}
83 1+(−1.44+6.31i)T+(−74.7−36.0i)T2 1 + (-1.44 + 6.31i)T + (-74.7 - 36.0i)T^{2}
89 1+(−10.2−12.9i)T+(−19.8+86.7i)T2 1 + (-10.2 - 12.9i)T + (-19.8 + 86.7i)T^{2}
97 1+(2.23−9.80i)T+(−87.3−42.0i)T2 1 + (2.23 - 9.80i)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

−13.33328786289006017623822044689, −12.26080821337876697931680632070, −11.52254472677245687643893826720, −10.39695854605230818572181306643, −9.400053005328048438913470314270, −8.580606293894004414992790591931, −7.39727395962489050433871042774, −4.59451425935125874738159811682, −3.62433848994998155292623325916, −0.840602669496891280351210769258, 3.18949509739808649516602472934, 6.01415881187197505451899146134, 6.76954759821916524734056361812, 7.82421778983140250247807195449, 8.672680076593385096086781910154, 9.931916942921962243781696739186, 11.52106907446574385740040894749, 12.22238145556166435774378668696, 14.27757045370274437708462084493, 14.84494962934818121891519417654

Graph of the ZZ-function along the critical line