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Properties

Label 2-87-29.25-c1-0-0
Degree 22
Conductor 8787
Sign −0.990+0.138i-0.990 + 0.138i
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  + (−0.563 + 2.47i)2-s + (−0.900 + 0.433i)3-s + (−3.98 − 1.91i)4-s + (−0.242 + 1.06i)5-s + (−0.563 − 2.47i)6-s + (−1.55 + 0.750i)7-s + (3.82 − 4.79i)8-s + (0.623 − 0.781i)9-s + (−2.48 − 1.19i)10-s + (3.92 + 4.92i)11-s + 4.42·12-s + (−0.797 − 1.00i)13-s + (−0.975 − 4.27i)14-s + (−0.242 − 1.06i)15-s + (4.18 + 5.24i)16-s − 5.08·17-s + ⋯
L(s)  = 1  + (−0.398 + 1.74i)2-s + (−0.520 + 0.250i)3-s + (−1.99 − 0.959i)4-s + (−0.108 + 0.475i)5-s + (−0.230 − 1.00i)6-s + (−0.589 + 0.283i)7-s + (1.35 − 1.69i)8-s + (0.207 − 0.260i)9-s + (−0.786 − 0.378i)10-s + (1.18 + 1.48i)11-s + 1.27·12-s + (−0.221 − 0.277i)13-s + (−0.260 − 1.14i)14-s + (−0.0625 − 0.274i)15-s + (1.04 + 1.31i)16-s − 1.23·17-s + ⋯

Functional equation

Λ(s)=(87s/2 ΓC(s) L(s)=((−0.990+0.138i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(87s/2 ΓC(s+1/2) L(s)=((−0.990+0.138i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8787    =    3⋅293 \cdot 29
Sign: −0.990+0.138i-0.990 + 0.138i
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), −0.990+0.138i)(2,\ 87,\ (\ :1/2),\ -0.990 + 0.138i)

Particular Values

L(1)L(1) ≈\approx 0.0392087−0.564053i0.0392087 - 0.564053i
L(12)L(\frac12) ≈\approx 0.0392087−0.564053i0.0392087 - 0.564053i
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+(1.89+5.04i)T 1 + (1.89 + 5.04i)T
good2 1+(0.563−2.47i)T+(−1.80−0.867i)T2 1 + (0.563 - 2.47i)T + (-1.80 - 0.867i)T^{2}
5 1+(0.242−1.06i)T+(−4.50−2.16i)T2 1 + (0.242 - 1.06i)T + (-4.50 - 2.16i)T^{2}
7 1+(1.55−0.750i)T+(4.36−5.47i)T2 1 + (1.55 - 0.750i)T + (4.36 - 5.47i)T^{2}
11 1+(−3.92−4.92i)T+(−2.44+10.7i)T2 1 + (-3.92 - 4.92i)T + (-2.44 + 10.7i)T^{2}
13 1+(0.797+1.00i)T+(−2.89+12.6i)T2 1 + (0.797 + 1.00i)T + (-2.89 + 12.6i)T^{2}
17 1+5.08T+17T2 1 + 5.08T + 17T^{2}
19 1+(−5.88−2.83i)T+(11.8+14.8i)T2 1 + (-5.88 - 2.83i)T + (11.8 + 14.8i)T^{2}
23 1+(−1.00−4.41i)T+(−20.7+9.97i)T2 1 + (-1.00 - 4.41i)T + (-20.7 + 9.97i)T^{2}
31 1+(−1.52+6.68i)T+(−27.9−13.4i)T2 1 + (-1.52 + 6.68i)T + (-27.9 - 13.4i)T^{2}
37 1+(−3.84+4.82i)T+(−8.23−36.0i)T2 1 + (-3.84 + 4.82i)T + (-8.23 - 36.0i)T^{2}
41 1−4.04T+41T2 1 - 4.04T + 41T^{2}
43 1+(0.407+1.78i)T+(−38.7+18.6i)T2 1 + (0.407 + 1.78i)T + (-38.7 + 18.6i)T^{2}
47 1+(−0.945−1.18i)T+(−10.4+45.8i)T2 1 + (-0.945 - 1.18i)T + (-10.4 + 45.8i)T^{2}
53 1+(−0.839+3.67i)T+(−47.7−22.9i)T2 1 + (-0.839 + 3.67i)T + (-47.7 - 22.9i)T^{2}
59 1+8.72T+59T2 1 + 8.72T + 59T^{2}
61 1+(−2.78+1.33i)T+(38.0−47.6i)T2 1 + (-2.78 + 1.33i)T + (38.0 - 47.6i)T^{2}
67 1+(1.49−1.87i)T+(−14.9−65.3i)T2 1 + (1.49 - 1.87i)T + (-14.9 - 65.3i)T^{2}
71 1+(−5.82−7.30i)T+(−15.7+69.2i)T2 1 + (-5.82 - 7.30i)T + (-15.7 + 69.2i)T^{2}
73 1+(−0.400−1.75i)T+(−65.7+31.6i)T2 1 + (-0.400 - 1.75i)T + (-65.7 + 31.6i)T^{2}
79 1+(1.78−2.24i)T+(−17.5−77.0i)T2 1 + (1.78 - 2.24i)T + (-17.5 - 77.0i)T^{2}
83 1+(5.94+2.86i)T+(51.7+64.8i)T2 1 + (5.94 + 2.86i)T + (51.7 + 64.8i)T^{2}
89 1+(0.744−3.26i)T+(−80.1−38.6i)T2 1 + (0.744 - 3.26i)T + (-80.1 - 38.6i)T^{2}
97 1+(2.04+0.983i)T+(60.4+75.8i)T2 1 + (2.04 + 0.983i)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

−15.16118050022843159921866083262, −14.19906347700537839649856219169, −12.87310152589461963112684182517, −11.50482433126427757286091545137, −9.742431912517094940363077985279, −9.324624227901577572686792548341, −7.55885984262009143855116758703, −6.77466013218414382650062451083, −5.73919089011354904261719852341, −4.30223817013189553533458349916, 0.938251382334138438353262608340, 3.17500430035784360813652181235, 4.64437198777814347121229335412, 6.63001479973554901091548056054, 8.648553435709736852854558542037, 9.325531234427143624171984320365, 10.70883349119752888891978463045, 11.43112893218702336535556728576, 12.28698186215596000945584962041, 13.23732712008420936976416394457

Graph of the ZZ-function along the critical line