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Properties

Label 2-87-29.20-c1-0-3
Degree 22
Conductor 8787
Sign 0.980−0.198i0.980 - 0.198i
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  + (0.626 − 0.301i)2-s + (0.623 + 0.781i)3-s + (−0.945 + 1.18i)4-s + (1.81 − 0.875i)5-s + (0.626 + 0.301i)6-s + (−1.49 − 1.87i)7-s + (−0.543 + 2.38i)8-s + (−0.222 + 0.974i)9-s + (0.874 − 1.09i)10-s + (0.213 + 0.933i)11-s − 1.51·12-s + (−1.45 − 6.36i)13-s + (−1.49 − 0.722i)14-s + (1.81 + 0.875i)15-s + (−0.297 − 1.30i)16-s − 3.81·17-s + ⋯
L(s)  = 1  + (0.442 − 0.213i)2-s + (0.359 + 0.451i)3-s + (−0.472 + 0.593i)4-s + (0.813 − 0.391i)5-s + (0.255 + 0.123i)6-s + (−0.564 − 0.707i)7-s + (−0.192 + 0.842i)8-s + (−0.0741 + 0.324i)9-s + (0.276 − 0.346i)10-s + (0.0642 + 0.281i)11-s − 0.437·12-s + (−0.403 − 1.76i)13-s + (−0.400 − 0.192i)14-s + (0.469 + 0.226i)15-s + (−0.0742 − 0.325i)16-s − 0.925·17-s + ⋯

Functional equation

Λ(s)=(87s/2 ΓC(s) L(s)=((0.980−0.198i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(87s/2 ΓC(s+1/2) L(s)=((0.980−0.198i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8787    =    3⋅293 \cdot 29
Sign: 0.980−0.198i0.980 - 0.198i
Analytic conductor: 0.6946980.694698
Root analytic conductor: 0.8334850.833485
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ87(49,⋅)\chi_{87} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 87, ( :1/2), 0.980−0.198i)(2,\ 87,\ (\ :1/2),\ 0.980 - 0.198i)

Particular Values

L(1)L(1) ≈\approx 1.21946+0.122332i1.21946 + 0.122332i
L(12)L(\frac12) ≈\approx 1.21946+0.122332i1.21946 + 0.122332i
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+(−0.623−0.781i)T 1 + (-0.623 - 0.781i)T
29 1+(−5.32−0.822i)T 1 + (-5.32 - 0.822i)T
good2 1+(−0.626+0.301i)T+(1.24−1.56i)T2 1 + (-0.626 + 0.301i)T + (1.24 - 1.56i)T^{2}
5 1+(−1.81+0.875i)T+(3.11−3.90i)T2 1 + (-1.81 + 0.875i)T + (3.11 - 3.90i)T^{2}
7 1+(1.49+1.87i)T+(−1.55+6.82i)T2 1 + (1.49 + 1.87i)T + (-1.55 + 6.82i)T^{2}
11 1+(−0.213−0.933i)T+(−9.91+4.77i)T2 1 + (-0.213 - 0.933i)T + (-9.91 + 4.77i)T^{2}
13 1+(1.45+6.36i)T+(−11.7+5.64i)T2 1 + (1.45 + 6.36i)T + (-11.7 + 5.64i)T^{2}
17 1+3.81T+17T2 1 + 3.81T + 17T^{2}
19 1+(2.69−3.37i)T+(−4.22−18.5i)T2 1 + (2.69 - 3.37i)T + (-4.22 - 18.5i)T^{2}
23 1+(−4.85−2.33i)T+(14.3+17.9i)T2 1 + (-4.85 - 2.33i)T + (14.3 + 17.9i)T^{2}
31 1+(2.46−1.18i)T+(19.3−24.2i)T2 1 + (2.46 - 1.18i)T + (19.3 - 24.2i)T^{2}
37 1+(−0.414+1.81i)T+(−33.3−16.0i)T2 1 + (-0.414 + 1.81i)T + (-33.3 - 16.0i)T^{2}
41 1−11.8T+41T2 1 - 11.8T + 41T^{2}
43 1+(−3.33−1.60i)T+(26.8+33.6i)T2 1 + (-3.33 - 1.60i)T + (26.8 + 33.6i)T^{2}
47 1+(−0.719−3.15i)T+(−42.3+20.3i)T2 1 + (-0.719 - 3.15i)T + (-42.3 + 20.3i)T^{2}
53 1+(2.04−0.983i)T+(33.0−41.4i)T2 1 + (2.04 - 0.983i)T + (33.0 - 41.4i)T^{2}
59 1+9.30T+59T2 1 + 9.30T + 59T^{2}
61 1+(−1.95−2.45i)T+(−13.5+59.4i)T2 1 + (-1.95 - 2.45i)T + (-13.5 + 59.4i)T^{2}
67 1+(2.69−11.7i)T+(−60.3−29.0i)T2 1 + (2.69 - 11.7i)T + (-60.3 - 29.0i)T^{2}
71 1+(−1.02−4.48i)T+(−63.9+30.8i)T2 1 + (-1.02 - 4.48i)T + (-63.9 + 30.8i)T^{2}
73 1+(8.19+3.94i)T+(45.5+57.0i)T2 1 + (8.19 + 3.94i)T + (45.5 + 57.0i)T^{2}
79 1+(0.954−4.18i)T+(−71.1−34.2i)T2 1 + (0.954 - 4.18i)T + (-71.1 - 34.2i)T^{2}
83 1+(−10.0+12.6i)T+(−18.4−80.9i)T2 1 + (-10.0 + 12.6i)T + (-18.4 - 80.9i)T^{2}
89 1+(13.9−6.70i)T+(55.4−69.5i)T2 1 + (13.9 - 6.70i)T + (55.4 - 69.5i)T^{2}
97 1+(−9.82+12.3i)T+(−21.5−94.5i)T2 1 + (-9.82 + 12.3i)T + (-21.5 - 94.5i)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

−14.00686756212010650108514069515, −13.04551241911722362111960096692, −12.66241706770323601678831358789, −10.84394895185136316765051470202, −9.821714634851823745537148712805, −8.802768864765464308260464492586, −7.53980768272226008651777164328, −5.63910136569424856062767399787, −4.35143892212928684892717348392, −2.93429556732389874933229058579, 2.38999204025373528884186425916, 4.52827333859278540616598365948, 6.17476363171916735581843429900, 6.74480056948357067830448435816, 8.990600444375428211614297167666, 9.403874665916473964606072038548, 10.89192969477129782820064787031, 12.38334003953773202386472265489, 13.39249347464210750220078345830, 14.06819514434524724865526330611

Graph of the ZZ-function along the critical line