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Properties

Label 2-800-100.23-c1-0-22
Degree 22
Conductor 800800
Sign 0.331+0.943i0.331 + 0.943i
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  + (0.366 + 0.719i)3-s + (−0.581 − 2.15i)5-s + (1.82 + 1.82i)7-s + (1.37 − 1.89i)9-s + (−2.56 − 3.52i)11-s + (−0.435 − 2.74i)13-s + (1.34 − 1.21i)15-s + (−3.87 − 1.97i)17-s + (−0.914 − 2.81i)19-s + (−0.642 + 1.97i)21-s + (−0.658 + 4.15i)23-s + (−4.32 + 2.51i)25-s + (4.26 + 0.675i)27-s + (1.54 + 0.503i)29-s + (10.0 − 3.25i)31-s + ⋯
L(s)  = 1  + (0.211 + 0.415i)3-s + (−0.260 − 0.965i)5-s + (0.688 + 0.688i)7-s + (0.459 − 0.633i)9-s + (−0.772 − 1.06i)11-s + (−0.120 − 0.762i)13-s + (0.346 − 0.312i)15-s + (−0.940 − 0.479i)17-s + (−0.209 − 0.645i)19-s + (−0.140 + 0.431i)21-s + (−0.137 + 0.866i)23-s + (−0.864 + 0.502i)25-s + (0.821 + 0.130i)27-s + (0.287 + 0.0934i)29-s + (1.79 − 0.583i)31-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.331+0.943i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+1/2) L(s)=((0.331+0.943i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.331+0.943i0.331 + 0.943i
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), 0.331+0.943i)(2,\ 800,\ (\ :1/2),\ 0.331 + 0.943i)

Particular Values

L(1)L(1) ≈\approx 1.20190−0.851603i1.20190 - 0.851603i
L(12)L(\frac12) ≈\approx 1.20190−0.851603i1.20190 - 0.851603i
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+(0.581+2.15i)T 1 + (0.581 + 2.15i)T
good3 1+(−0.366−0.719i)T+(−1.76+2.42i)T2 1 + (-0.366 - 0.719i)T + (-1.76 + 2.42i)T^{2}
7 1+(−1.82−1.82i)T+7iT2 1 + (-1.82 - 1.82i)T + 7iT^{2}
11 1+(2.56+3.52i)T+(−3.39+10.4i)T2 1 + (2.56 + 3.52i)T + (-3.39 + 10.4i)T^{2}
13 1+(0.435+2.74i)T+(−12.3+4.01i)T2 1 + (0.435 + 2.74i)T + (-12.3 + 4.01i)T^{2}
17 1+(3.87+1.97i)T+(9.99+13.7i)T2 1 + (3.87 + 1.97i)T + (9.99 + 13.7i)T^{2}
19 1+(0.914+2.81i)T+(−15.3+11.1i)T2 1 + (0.914 + 2.81i)T + (-15.3 + 11.1i)T^{2}
23 1+(0.658−4.15i)T+(−21.8−7.10i)T2 1 + (0.658 - 4.15i)T + (-21.8 - 7.10i)T^{2}
29 1+(−1.54−0.503i)T+(23.4+17.0i)T2 1 + (-1.54 - 0.503i)T + (23.4 + 17.0i)T^{2}
31 1+(−10.0+3.25i)T+(25.0−18.2i)T2 1 + (-10.0 + 3.25i)T + (25.0 - 18.2i)T^{2}
37 1+(−3.29+0.521i)T+(35.1−11.4i)T2 1 + (-3.29 + 0.521i)T + (35.1 - 11.4i)T^{2}
41 1+(8.06+5.86i)T+(12.6+38.9i)T2 1 + (8.06 + 5.86i)T + (12.6 + 38.9i)T^{2}
43 1+(−2.01+2.01i)T−43iT2 1 + (-2.01 + 2.01i)T - 43iT^{2}
47 1+(0.681−0.347i)T+(27.6−38.0i)T2 1 + (0.681 - 0.347i)T + (27.6 - 38.0i)T^{2}
53 1+(4.65−2.37i)T+(31.1−42.8i)T2 1 + (4.65 - 2.37i)T + (31.1 - 42.8i)T^{2}
59 1+(0.111+0.0810i)T+(18.2+56.1i)T2 1 + (0.111 + 0.0810i)T + (18.2 + 56.1i)T^{2}
61 1+(−9.77+7.10i)T+(18.8−58.0i)T2 1 + (-9.77 + 7.10i)T + (18.8 - 58.0i)T^{2}
67 1+(−2.64+5.19i)T+(−39.3−54.2i)T2 1 + (-2.64 + 5.19i)T + (-39.3 - 54.2i)T^{2}
71 1+(0.559+0.181i)T+(57.4+41.7i)T2 1 + (0.559 + 0.181i)T + (57.4 + 41.7i)T^{2}
73 1+(−15.0−2.38i)T+(69.4+22.5i)T2 1 + (-15.0 - 2.38i)T + (69.4 + 22.5i)T^{2}
79 1+(3.31−10.1i)T+(−63.9−46.4i)T2 1 + (3.31 - 10.1i)T + (-63.9 - 46.4i)T^{2}
83 1+(−0.0483−0.0246i)T+(48.7+67.1i)T2 1 + (-0.0483 - 0.0246i)T + (48.7 + 67.1i)T^{2}
89 1+(−8.88−12.2i)T+(−27.5+84.6i)T2 1 + (-8.88 - 12.2i)T + (-27.5 + 84.6i)T^{2}
97 1+(5.19+10.1i)T+(−57.0+78.4i)T2 1 + (5.19 + 10.1i)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

−9.957026346449480323423639903259, −9.140658471616190965786231161233, −8.437653247536228011419079026986, −7.896947671611974467857865070937, −6.53678243877993749032021096670, −5.37136402367076535630046124242, −4.82574623824258095186532461805, −3.68829667992421747797019336709, −2.46499658894099140313498655818, −0.72819696122503841075720534881, 1.75902572221140903975564417175, 2.64205415895498533742796837960, 4.26403450524420778163202299313, 4.73554558967451093186345424433, 6.46397098995040338683171626477, 6.98573712179029866807075097418, 7.87186867996742881365570982009, 8.346820739928880187188396258234, 9.962136721786674688120892214011, 10.36377496682543410823898965911

Graph of the ZZ-function along the critical line