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Properties

Label 2-800-100.23-c1-0-11
Degree 22
Conductor 800800
Sign 0.928+0.372i0.928 + 0.372i
Analytic cond. 6.388036.38803
Root an. cond. 2.527452.52745
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  + (−0.927 − 1.81i)3-s + (2.22 + 0.215i)5-s + (−0.168 − 0.168i)7-s + (−0.687 + 0.946i)9-s + (3.26 + 4.49i)11-s + (0.749 + 4.73i)13-s + (−1.67 − 4.24i)15-s + (6.52 + 3.32i)17-s + (−2.15 − 6.61i)19-s + (−0.150 + 0.461i)21-s + (−0.984 + 6.21i)23-s + (4.90 + 0.958i)25-s + (−3.69 − 0.584i)27-s + (−0.403 − 0.131i)29-s + (6.52 − 2.12i)31-s + ⋯
L(s)  = 1  + (−0.535 − 1.05i)3-s + (0.995 + 0.0963i)5-s + (−0.0635 − 0.0635i)7-s + (−0.229 + 0.315i)9-s + (0.985 + 1.35i)11-s + (0.207 + 1.31i)13-s + (−0.431 − 1.09i)15-s + (1.58 + 0.805i)17-s + (−0.493 − 1.51i)19-s + (−0.0327 + 0.100i)21-s + (−0.205 + 1.29i)23-s + (0.981 + 0.191i)25-s + (−0.710 − 0.112i)27-s + (−0.0749 − 0.0243i)29-s + (1.17 − 0.380i)31-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.928+0.372i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+1/2) L(s)=((0.928+0.372i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.928+0.372i0.928 + 0.372i
Analytic conductor: 6.388036.38803
Root analytic conductor: 2.527452.52745
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ800(223,⋅)\chi_{800} (223, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 800, ( :1/2), 0.928+0.372i)(2,\ 800,\ (\ :1/2),\ 0.928 + 0.372i)

Particular Values

L(1)L(1) ≈\approx 1.64349−0.317300i1.64349 - 0.317300i
L(12)L(\frac12) ≈\approx 1.64349−0.317300i1.64349 - 0.317300i
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)
bad2 1 1
5 1+(−2.22−0.215i)T 1 + (-2.22 - 0.215i)T
good3 1+(0.927+1.81i)T+(−1.76+2.42i)T2 1 + (0.927 + 1.81i)T + (-1.76 + 2.42i)T^{2}
7 1+(0.168+0.168i)T+7iT2 1 + (0.168 + 0.168i)T + 7iT^{2}
11 1+(−3.26−4.49i)T+(−3.39+10.4i)T2 1 + (-3.26 - 4.49i)T + (-3.39 + 10.4i)T^{2}
13 1+(−0.749−4.73i)T+(−12.3+4.01i)T2 1 + (-0.749 - 4.73i)T + (-12.3 + 4.01i)T^{2}
17 1+(−6.52−3.32i)T+(9.99+13.7i)T2 1 + (-6.52 - 3.32i)T + (9.99 + 13.7i)T^{2}
19 1+(2.15+6.61i)T+(−15.3+11.1i)T2 1 + (2.15 + 6.61i)T + (-15.3 + 11.1i)T^{2}
23 1+(0.984−6.21i)T+(−21.8−7.10i)T2 1 + (0.984 - 6.21i)T + (-21.8 - 7.10i)T^{2}
29 1+(0.403+0.131i)T+(23.4+17.0i)T2 1 + (0.403 + 0.131i)T + (23.4 + 17.0i)T^{2}
31 1+(−6.52+2.12i)T+(25.0−18.2i)T2 1 + (-6.52 + 2.12i)T + (25.0 - 18.2i)T^{2}
37 1+(10.6−1.68i)T+(35.1−11.4i)T2 1 + (10.6 - 1.68i)T + (35.1 - 11.4i)T^{2}
41 1+(2.78+2.02i)T+(12.6+38.9i)T2 1 + (2.78 + 2.02i)T + (12.6 + 38.9i)T^{2}
43 1+(−0.0127+0.0127i)T−43iT2 1 + (-0.0127 + 0.0127i)T - 43iT^{2}
47 1+(−3.27+1.66i)T+(27.6−38.0i)T2 1 + (-3.27 + 1.66i)T + (27.6 - 38.0i)T^{2}
53 1+(−3.62+1.84i)T+(31.1−42.8i)T2 1 + (-3.62 + 1.84i)T + (31.1 - 42.8i)T^{2}
59 1+(5.38+3.90i)T+(18.2+56.1i)T2 1 + (5.38 + 3.90i)T + (18.2 + 56.1i)T^{2}
61 1+(−7.47+5.43i)T+(18.8−58.0i)T2 1 + (-7.47 + 5.43i)T + (18.8 - 58.0i)T^{2}
67 1+(1.62−3.19i)T+(−39.3−54.2i)T2 1 + (1.62 - 3.19i)T + (-39.3 - 54.2i)T^{2}
71 1+(−5.30−1.72i)T+(57.4+41.7i)T2 1 + (-5.30 - 1.72i)T + (57.4 + 41.7i)T^{2}
73 1+(−3.26−0.517i)T+(69.4+22.5i)T2 1 + (-3.26 - 0.517i)T + (69.4 + 22.5i)T^{2}
79 1+(−0.104+0.321i)T+(−63.9−46.4i)T2 1 + (-0.104 + 0.321i)T + (-63.9 - 46.4i)T^{2}
83 1+(−2.12−1.08i)T+(48.7+67.1i)T2 1 + (-2.12 - 1.08i)T + (48.7 + 67.1i)T^{2}
89 1+(5.36+7.38i)T+(−27.5+84.6i)T2 1 + (5.36 + 7.38i)T + (-27.5 + 84.6i)T^{2}
97 1+(8.01+15.7i)T+(−57.0+78.4i)T2 1 + (8.01 + 15.7i)T + (-57.0 + 78.4i)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

−9.991013467764307321403970771020, −9.561672999594277807749299110171, −8.558281377639165762041412757122, −7.13623670460859759579149390357, −6.86122343467843283759562892258, −6.05539652734845502101888789512, −5.03452377919477482524761325956, −3.79228364728266145775791055544, −2.03024454039235556984425698157, −1.38564027025147450452559782658, 1.10117401263796839007631004478, 2.97631927667734643751343217702, 3.89794344356674081968704122628, 5.20853256649344487371504904424, 5.72803273249248838088747445034, 6.44254683081575866875557499908, 7.979779552847490036947343157645, 8.765765456195295839416107050559, 9.717187108786917793261517333580, 10.40095545599376447279197069298

Graph of the ZZ-function along the critical line