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Properties

Label 2-87-29.25-c1-0-5
Degree 22
Conductor 8787
Sign −0.543+0.839i-0.543 + 0.839i
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  + (0.596 − 2.61i)2-s + (0.900 − 0.433i)3-s + (−4.67 − 2.25i)4-s + (−0.692 + 3.03i)5-s + (−0.596 − 2.61i)6-s + (2.13 − 1.02i)7-s + (−5.32 + 6.68i)8-s + (0.623 − 0.781i)9-s + (7.51 + 3.61i)10-s + (1.62 + 2.03i)11-s − 5.18·12-s + (−1.43 − 1.80i)13-s + (−1.41 − 6.19i)14-s + (0.692 + 3.03i)15-s + (7.81 + 9.80i)16-s − 4.83·17-s + ⋯
L(s)  = 1  + (0.421 − 1.84i)2-s + (0.520 − 0.250i)3-s + (−2.33 − 1.12i)4-s + (−0.309 + 1.35i)5-s + (−0.243 − 1.06i)6-s + (0.807 − 0.388i)7-s + (−1.88 + 2.36i)8-s + (0.207 − 0.260i)9-s + (2.37 + 1.14i)10-s + (0.489 + 0.613i)11-s − 1.49·12-s + (−0.399 − 0.500i)13-s + (−0.377 − 1.65i)14-s + (0.178 + 0.783i)15-s + (1.95 + 2.45i)16-s − 1.17·17-s + ⋯

Functional equation

Λ(s)=(87s/2 ΓC(s) L(s)=((−0.543+0.839i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(87s/2 ΓC(s+1/2) L(s)=((−0.543+0.839i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8787    =    3⋅293 \cdot 29
Sign: −0.543+0.839i-0.543 + 0.839i
Analytic conductor: 0.6946980.694698
Root analytic conductor: 0.8334850.833485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ87(25,⋅)\chi_{87} (25, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 87, ( :1/2), −0.543+0.839i)(2,\ 87,\ (\ :1/2),\ -0.543 + 0.839i)

Particular Values

L(1)L(1) ≈\approx 0.567605−1.04317i0.567605 - 1.04317i
L(12)L(\frac12) ≈\approx 0.567605−1.04317i0.567605 - 1.04317i
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+(−5.05−1.86i)T 1 + (-5.05 - 1.86i)T
good2 1+(−0.596+2.61i)T+(−1.80−0.867i)T2 1 + (-0.596 + 2.61i)T + (-1.80 - 0.867i)T^{2}
5 1+(0.692−3.03i)T+(−4.50−2.16i)T2 1 + (0.692 - 3.03i)T + (-4.50 - 2.16i)T^{2}
7 1+(−2.13+1.02i)T+(4.36−5.47i)T2 1 + (-2.13 + 1.02i)T + (4.36 - 5.47i)T^{2}
11 1+(−1.62−2.03i)T+(−2.44+10.7i)T2 1 + (-1.62 - 2.03i)T + (-2.44 + 10.7i)T^{2}
13 1+(1.43+1.80i)T+(−2.89+12.6i)T2 1 + (1.43 + 1.80i)T + (-2.89 + 12.6i)T^{2}
17 1+4.83T+17T2 1 + 4.83T + 17T^{2}
19 1+(1.47+0.710i)T+(11.8+14.8i)T2 1 + (1.47 + 0.710i)T + (11.8 + 14.8i)T^{2}
23 1+(−0.263−1.15i)T+(−20.7+9.97i)T2 1 + (-0.263 - 1.15i)T + (-20.7 + 9.97i)T^{2}
31 1+(1.28−5.63i)T+(−27.9−13.4i)T2 1 + (1.28 - 5.63i)T + (-27.9 - 13.4i)T^{2}
37 1+(−4.30+5.39i)T+(−8.23−36.0i)T2 1 + (-4.30 + 5.39i)T + (-8.23 - 36.0i)T^{2}
41 1+7.66T+41T2 1 + 7.66T + 41T^{2}
43 1+(0.419+1.84i)T+(−38.7+18.6i)T2 1 + (0.419 + 1.84i)T + (-38.7 + 18.6i)T^{2}
47 1+(5.79+7.27i)T+(−10.4+45.8i)T2 1 + (5.79 + 7.27i)T + (-10.4 + 45.8i)T^{2}
53 1+(−1.15+5.04i)T+(−47.7−22.9i)T2 1 + (-1.15 + 5.04i)T + (-47.7 - 22.9i)T^{2}
59 1+4.90T+59T2 1 + 4.90T + 59T^{2}
61 1+(−10.3+5.00i)T+(38.0−47.6i)T2 1 + (-10.3 + 5.00i)T + (38.0 - 47.6i)T^{2}
67 1+(4.34−5.45i)T+(−14.9−65.3i)T2 1 + (4.34 - 5.45i)T + (-14.9 - 65.3i)T^{2}
71 1+(−2.32−2.92i)T+(−15.7+69.2i)T2 1 + (-2.32 - 2.92i)T + (-15.7 + 69.2i)T^{2}
73 1+(0.532+2.33i)T+(−65.7+31.6i)T2 1 + (0.532 + 2.33i)T + (-65.7 + 31.6i)T^{2}
79 1+(−2.34+2.93i)T+(−17.5−77.0i)T2 1 + (-2.34 + 2.93i)T + (-17.5 - 77.0i)T^{2}
83 1+(−9.54−4.59i)T+(51.7+64.8i)T2 1 + (-9.54 - 4.59i)T + (51.7 + 64.8i)T^{2}
89 1+(−1.26+5.55i)T+(−80.1−38.6i)T2 1 + (-1.26 + 5.55i)T + (-80.1 - 38.6i)T^{2}
97 1+(15.1+7.28i)T+(60.4+75.8i)T2 1 + (15.1 + 7.28i)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

−13.73437766346699013638322099364, −12.62707780425592769190633463340, −11.53627588225444625585037759735, −10.80094571347772997344294808152, −9.937692601084918336223512194282, −8.547074224549717038125896281475, −6.95977393484443984075993531816, −4.68060400702176963914707979818, −3.39878355710959014789584512321, −2.09730961705240444748345141327, 4.27790468720207800230934456506, 4.94289481757606924129502933645, 6.40708763929686860673538073046, 7.989095101302902548308098827481, 8.576070669138318105499200333447, 9.335428144864701899625525865974, 11.75114440425857549769520759626, 12.96174536832155259188761446268, 13.79229880352355306982404814772, 14.77663531250962499054413475817

Graph of the ZZ-function along the critical line