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Properties

Label 2-800-100.23-c1-0-1
Degree 22
Conductor 800800
Sign 0.761−0.648i0.761 - 0.648i
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  + (−1.18 − 2.32i)3-s + (0.966 + 2.01i)5-s + (0.0730 + 0.0730i)7-s + (−2.24 + 3.09i)9-s + (−0.496 − 0.683i)11-s + (0.983 + 6.20i)13-s + (3.54 − 4.64i)15-s + (−4.79 − 2.44i)17-s + (2.38 + 7.33i)19-s + (0.0833 − 0.256i)21-s + (−0.254 + 1.60i)23-s + (−3.13 + 3.89i)25-s + (2.13 + 0.337i)27-s + (8.34 + 2.71i)29-s + (−6.65 + 2.16i)31-s + ⋯
L(s)  = 1  + (−0.684 − 1.34i)3-s + (0.432 + 0.901i)5-s + (0.0276 + 0.0276i)7-s + (−0.749 + 1.03i)9-s + (−0.149 − 0.206i)11-s + (0.272 + 1.72i)13-s + (0.915 − 1.19i)15-s + (−1.16 − 0.592i)17-s + (0.546 + 1.68i)19-s + (0.0181 − 0.0560i)21-s + (−0.0530 + 0.334i)23-s + (−0.626 + 0.779i)25-s + (0.410 + 0.0650i)27-s + (1.55 + 0.503i)29-s + (−1.19 + 0.388i)31-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.761−0.648i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+1/2) L(s)=((0.761−0.648i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.761−0.648i0.761 - 0.648i
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), 0.761−0.648i)(2,\ 800,\ (\ :1/2),\ 0.761 - 0.648i)

Particular Values

L(1)L(1) ≈\approx 0.984587+0.362540i0.984587 + 0.362540i
L(12)L(\frac12) ≈\approx 0.984587+0.362540i0.984587 + 0.362540i
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+(−0.966−2.01i)T 1 + (-0.966 - 2.01i)T
good3 1+(1.18+2.32i)T+(−1.76+2.42i)T2 1 + (1.18 + 2.32i)T + (-1.76 + 2.42i)T^{2}
7 1+(−0.0730−0.0730i)T+7iT2 1 + (-0.0730 - 0.0730i)T + 7iT^{2}
11 1+(0.496+0.683i)T+(−3.39+10.4i)T2 1 + (0.496 + 0.683i)T + (-3.39 + 10.4i)T^{2}
13 1+(−0.983−6.20i)T+(−12.3+4.01i)T2 1 + (-0.983 - 6.20i)T + (-12.3 + 4.01i)T^{2}
17 1+(4.79+2.44i)T+(9.99+13.7i)T2 1 + (4.79 + 2.44i)T + (9.99 + 13.7i)T^{2}
19 1+(−2.38−7.33i)T+(−15.3+11.1i)T2 1 + (-2.38 - 7.33i)T + (-15.3 + 11.1i)T^{2}
23 1+(0.254−1.60i)T+(−21.8−7.10i)T2 1 + (0.254 - 1.60i)T + (-21.8 - 7.10i)T^{2}
29 1+(−8.34−2.71i)T+(23.4+17.0i)T2 1 + (-8.34 - 2.71i)T + (23.4 + 17.0i)T^{2}
31 1+(6.65−2.16i)T+(25.0−18.2i)T2 1 + (6.65 - 2.16i)T + (25.0 - 18.2i)T^{2}
37 1+(0.0262−0.00416i)T+(35.1−11.4i)T2 1 + (0.0262 - 0.00416i)T + (35.1 - 11.4i)T^{2}
41 1+(−4.26−3.09i)T+(12.6+38.9i)T2 1 + (-4.26 - 3.09i)T + (12.6 + 38.9i)T^{2}
43 1+(−1.83+1.83i)T−43iT2 1 + (-1.83 + 1.83i)T - 43iT^{2}
47 1+(−5.86+2.99i)T+(27.6−38.0i)T2 1 + (-5.86 + 2.99i)T + (27.6 - 38.0i)T^{2}
53 1+(−0.916+0.467i)T+(31.1−42.8i)T2 1 + (-0.916 + 0.467i)T + (31.1 - 42.8i)T^{2}
59 1+(−5.94−4.32i)T+(18.2+56.1i)T2 1 + (-5.94 - 4.32i)T + (18.2 + 56.1i)T^{2}
61 1+(1.59−1.15i)T+(18.8−58.0i)T2 1 + (1.59 - 1.15i)T + (18.8 - 58.0i)T^{2}
67 1+(0.166−0.326i)T+(−39.3−54.2i)T2 1 + (0.166 - 0.326i)T + (-39.3 - 54.2i)T^{2}
71 1+(−6.56−2.13i)T+(57.4+41.7i)T2 1 + (-6.56 - 2.13i)T + (57.4 + 41.7i)T^{2}
73 1+(0.655+0.103i)T+(69.4+22.5i)T2 1 + (0.655 + 0.103i)T + (69.4 + 22.5i)T^{2}
79 1+(1.26−3.88i)T+(−63.9−46.4i)T2 1 + (1.26 - 3.88i)T + (-63.9 - 46.4i)T^{2}
83 1+(6.93+3.53i)T+(48.7+67.1i)T2 1 + (6.93 + 3.53i)T + (48.7 + 67.1i)T^{2}
89 1+(−9.44−12.9i)T+(−27.5+84.6i)T2 1 + (-9.44 - 12.9i)T + (-27.5 + 84.6i)T^{2}
97 1+(−1.11−2.19i)T+(−57.0+78.4i)T2 1 + (-1.11 - 2.19i)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

−10.57693952592304824468621349121, −9.546873762947249604693582293983, −8.564043571431006130191411217481, −7.40191198326252025794901220496, −6.84875499470089624132718409556, −6.24338548308869620059547150436, −5.37347784328672652133896374880, −3.88586121848673359853710959335, −2.39377897873565473613468174033, −1.48653673881592078767576635141, 0.58925407223284218165361835874, 2.65208838672684217451558249305, 4.07974226767765831828832462708, 4.83259347928931073022225422763, 5.48580118556515358812789547450, 6.32742647691056521644935178797, 7.76016316953778837494315410722, 8.808733981199198670621244998773, 9.354153187844043386299485038822, 10.30834286959137889228365802493

Graph of the ZZ-function along the critical line