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Properties

Label 2-87-29.25-c1-0-1
Degree 22
Conductor 8787
Sign 0.0372−0.999i0.0372 - 0.999i
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  + (−0.399 + 1.74i)2-s + (0.900 − 0.433i)3-s + (−1.10 − 0.529i)4-s + (−0.299 + 1.31i)5-s + (0.399 + 1.74i)6-s + (1.00 − 0.483i)7-s + (−0.871 + 1.09i)8-s + (0.623 − 0.781i)9-s + (−2.17 − 1.04i)10-s + (−1.53 − 1.93i)11-s − 1.22·12-s + (−1.75 − 2.20i)13-s + (0.444 + 1.94i)14-s + (0.299 + 1.31i)15-s + (−3.08 − 3.87i)16-s + 5.50·17-s + ⋯
L(s)  = 1  + (−0.282 + 1.23i)2-s + (0.520 − 0.250i)3-s + (−0.550 − 0.264i)4-s + (−0.133 + 0.586i)5-s + (0.163 + 0.714i)6-s + (0.379 − 0.182i)7-s + (−0.308 + 0.386i)8-s + (0.207 − 0.260i)9-s + (−0.687 − 0.331i)10-s + (−0.464 − 0.582i)11-s − 0.352·12-s + (−0.487 − 0.610i)13-s + (0.118 + 0.520i)14-s + (0.0772 + 0.338i)15-s + (−0.771 − 0.967i)16-s + 1.33·17-s + ⋯

Functional equation

Λ(s)=(87s/2 ΓC(s) L(s)=((0.0372−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0372 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(87s/2 ΓC(s+1/2) L(s)=((0.0372−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0372 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8787    =    3⋅293 \cdot 29
Sign: 0.0372−0.999i0.0372 - 0.999i
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), 0.0372−0.999i)(2,\ 87,\ (\ :1/2),\ 0.0372 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.709558+0.683571i0.709558 + 0.683571i
L(12)L(\frac12) ≈\approx 0.709558+0.683571i0.709558 + 0.683571i
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+(5.29−0.981i)T 1 + (5.29 - 0.981i)T
good2 1+(0.399−1.74i)T+(−1.80−0.867i)T2 1 + (0.399 - 1.74i)T + (-1.80 - 0.867i)T^{2}
5 1+(0.299−1.31i)T+(−4.50−2.16i)T2 1 + (0.299 - 1.31i)T + (-4.50 - 2.16i)T^{2}
7 1+(−1.00+0.483i)T+(4.36−5.47i)T2 1 + (-1.00 + 0.483i)T + (4.36 - 5.47i)T^{2}
11 1+(1.53+1.93i)T+(−2.44+10.7i)T2 1 + (1.53 + 1.93i)T + (-2.44 + 10.7i)T^{2}
13 1+(1.75+2.20i)T+(−2.89+12.6i)T2 1 + (1.75 + 2.20i)T + (-2.89 + 12.6i)T^{2}
17 1−5.50T+17T2 1 - 5.50T + 17T^{2}
19 1+(−0.318−0.153i)T+(11.8+14.8i)T2 1 + (-0.318 - 0.153i)T + (11.8 + 14.8i)T^{2}
23 1+(1.18+5.18i)T+(−20.7+9.97i)T2 1 + (1.18 + 5.18i)T + (-20.7 + 9.97i)T^{2}
31 1+(−0.990+4.33i)T+(−27.9−13.4i)T2 1 + (-0.990 + 4.33i)T + (-27.9 - 13.4i)T^{2}
37 1+(6.00−7.52i)T+(−8.23−36.0i)T2 1 + (6.00 - 7.52i)T + (-8.23 - 36.0i)T^{2}
41 1+9.05T+41T2 1 + 9.05T + 41T^{2}
43 1+(−2.01−8.82i)T+(−38.7+18.6i)T2 1 + (-2.01 - 8.82i)T + (-38.7 + 18.6i)T^{2}
47 1+(−2.24−2.81i)T+(−10.4+45.8i)T2 1 + (-2.24 - 2.81i)T + (-10.4 + 45.8i)T^{2}
53 1+(−1.46+6.41i)T+(−47.7−22.9i)T2 1 + (-1.46 + 6.41i)T + (-47.7 - 22.9i)T^{2}
59 1+6.26T+59T2 1 + 6.26T + 59T^{2}
61 1+(0.106−0.0512i)T+(38.0−47.6i)T2 1 + (0.106 - 0.0512i)T + (38.0 - 47.6i)T^{2}
67 1+(−8.91+11.1i)T+(−14.9−65.3i)T2 1 + (-8.91 + 11.1i)T + (-14.9 - 65.3i)T^{2}
71 1+(−4.68−5.86i)T+(−15.7+69.2i)T2 1 + (-4.68 - 5.86i)T + (-15.7 + 69.2i)T^{2}
73 1+(−3.54−15.5i)T+(−65.7+31.6i)T2 1 + (-3.54 - 15.5i)T + (-65.7 + 31.6i)T^{2}
79 1+(−0.160+0.200i)T+(−17.5−77.0i)T2 1 + (-0.160 + 0.200i)T + (-17.5 - 77.0i)T^{2}
83 1+(−5.52−2.65i)T+(51.7+64.8i)T2 1 + (-5.52 - 2.65i)T + (51.7 + 64.8i)T^{2}
89 1+(1.66−7.31i)T+(−80.1−38.6i)T2 1 + (1.66 - 7.31i)T + (-80.1 - 38.6i)T^{2}
97 1+(6.37+3.07i)T+(60.4+75.8i)T2 1 + (6.37 + 3.07i)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

−14.66609146321388820445480692163, −13.89833404375414612127693639093, −12.49862358696200197818924384935, −11.14166482605076096030060578330, −9.833408095384208736425607964526, −8.314037887750172186842692903276, −7.73673413998218321588103746075, −6.63102492328295062507112919443, −5.28678870161059551461664721660, −3.02280751128944763674251069583, 1.91113329301265944928779355277, 3.59486991487819349566625928273, 5.15520813837550531855458318983, 7.37124917646129727597585952061, 8.749281084830674980893654728294, 9.678802075328873372790333170407, 10.57970530246585255993190735620, 11.91049142222315017987804321054, 12.47299811079381710247712652747, 13.74349436034626932759294383639

Graph of the ZZ-function along the critical line