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Properties

Label 2-87-29.7-c1-0-5
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(7,⋅)\chi_{87} (7, \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

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

Graph of the ZZ-function along the critical line