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Properties

Label 2-10e3-25.19-c1-0-4
Degree 22
Conductor 10001000
Sign 0.999−0.0361i0.999 - 0.0361i
Analytic cond. 7.985047.98504
Root an. cond. 2.825782.82578
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  + (−0.919 − 1.26i)3-s − 0.0338i·7-s + (0.170 − 0.523i)9-s + (1.79 + 5.51i)11-s + (2.96 + 0.964i)13-s + (−2.40 + 3.30i)17-s + (−0.00789 − 0.00573i)19-s + (−0.0429 + 0.0311i)21-s + (−2.20 + 0.717i)23-s + (−5.28 + 1.71i)27-s + (4.38 − 3.18i)29-s + (3.80 + 2.76i)31-s + (5.33 − 7.33i)33-s + (10.1 + 3.29i)37-s + (−1.50 − 4.64i)39-s + ⋯
L(s)  = 1  + (−0.531 − 0.731i)3-s − 0.0128i·7-s + (0.0566 − 0.174i)9-s + (0.540 + 1.66i)11-s + (0.822 + 0.267i)13-s + (−0.583 + 0.802i)17-s + (−0.00181 − 0.00131i)19-s + (−0.00936 + 0.00680i)21-s + (−0.460 + 0.149i)23-s + (−1.01 + 0.330i)27-s + (0.814 − 0.592i)29-s + (0.682 + 0.495i)31-s + (0.928 − 1.27i)33-s + (1.66 + 0.541i)37-s + (−0.241 − 0.743i)39-s + ⋯

Functional equation

Λ(s)=(1000s/2 ΓC(s) L(s)=((0.999−0.0361i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0361i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1000s/2 ΓC(s+1/2) L(s)=((0.999−0.0361i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0361i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10001000    =    23⋅532^{3} \cdot 5^{3}
Sign: 0.999−0.0361i0.999 - 0.0361i
Analytic conductor: 7.985047.98504
Root analytic conductor: 2.825782.82578
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1000(849,⋅)\chi_{1000} (849, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1000, ( :1/2), 0.999−0.0361i)(2,\ 1000,\ (\ :1/2),\ 0.999 - 0.0361i)

Particular Values

L(1)L(1) ≈\approx 1.36110+0.0246241i1.36110 + 0.0246241i
L(12)L(\frac12) ≈\approx 1.36110+0.0246241i1.36110 + 0.0246241i
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 1
good3 1+(0.919+1.26i)T+(−0.927+2.85i)T2 1 + (0.919 + 1.26i)T + (-0.927 + 2.85i)T^{2}
7 1+0.0338iT−7T2 1 + 0.0338iT - 7T^{2}
11 1+(−1.79−5.51i)T+(−8.89+6.46i)T2 1 + (-1.79 - 5.51i)T + (-8.89 + 6.46i)T^{2}
13 1+(−2.96−0.964i)T+(10.5+7.64i)T2 1 + (-2.96 - 0.964i)T + (10.5 + 7.64i)T^{2}
17 1+(2.40−3.30i)T+(−5.25−16.1i)T2 1 + (2.40 - 3.30i)T + (-5.25 - 16.1i)T^{2}
19 1+(0.00789+0.00573i)T+(5.87+18.0i)T2 1 + (0.00789 + 0.00573i)T + (5.87 + 18.0i)T^{2}
23 1+(2.20−0.717i)T+(18.6−13.5i)T2 1 + (2.20 - 0.717i)T + (18.6 - 13.5i)T^{2}
29 1+(−4.38+3.18i)T+(8.96−27.5i)T2 1 + (-4.38 + 3.18i)T + (8.96 - 27.5i)T^{2}
31 1+(−3.80−2.76i)T+(9.57+29.4i)T2 1 + (-3.80 - 2.76i)T + (9.57 + 29.4i)T^{2}
37 1+(−10.1−3.29i)T+(29.9+21.7i)T2 1 + (-10.1 - 3.29i)T + (29.9 + 21.7i)T^{2}
41 1+(−1.81+5.59i)T+(−33.1−24.0i)T2 1 + (-1.81 + 5.59i)T + (-33.1 - 24.0i)T^{2}
43 1−0.480iT−43T2 1 - 0.480iT - 43T^{2}
47 1+(−6.66−9.17i)T+(−14.5+44.6i)T2 1 + (-6.66 - 9.17i)T + (-14.5 + 44.6i)T^{2}
53 1+(4.10+5.64i)T+(−16.3+50.4i)T2 1 + (4.10 + 5.64i)T + (-16.3 + 50.4i)T^{2}
59 1+(−2.19+6.74i)T+(−47.7−34.6i)T2 1 + (-2.19 + 6.74i)T + (-47.7 - 34.6i)T^{2}
61 1+(1.64+5.05i)T+(−49.3+35.8i)T2 1 + (1.64 + 5.05i)T + (-49.3 + 35.8i)T^{2}
67 1+(−1.71+2.36i)T+(−20.7−63.7i)T2 1 + (-1.71 + 2.36i)T + (-20.7 - 63.7i)T^{2}
71 1+(12.6−9.22i)T+(21.9−67.5i)T2 1 + (12.6 - 9.22i)T + (21.9 - 67.5i)T^{2}
73 1+(−6.82+2.21i)T+(59.0−42.9i)T2 1 + (-6.82 + 2.21i)T + (59.0 - 42.9i)T^{2}
79 1+(0.289−0.210i)T+(24.4−75.1i)T2 1 + (0.289 - 0.210i)T + (24.4 - 75.1i)T^{2}
83 1+(−1.89+2.60i)T+(−25.6−78.9i)T2 1 + (-1.89 + 2.60i)T + (-25.6 - 78.9i)T^{2}
89 1+(−4.71−14.4i)T+(−72.0+52.3i)T2 1 + (-4.71 - 14.4i)T + (-72.0 + 52.3i)T^{2}
97 1+(−1.26−1.73i)T+(−29.9+92.2i)T2 1 + (-1.26 - 1.73i)T + (-29.9 + 92.2i)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

−9.928266303432955045318269467045, −9.239745651337504503510905409547, −8.200874401688618982482102254632, −7.30804322959872063485659827196, −6.50435133557321436677401952622, −6.04360866129777074212599477042, −4.62087345130988271234856478964, −3.89672514993249295812856842657, −2.22110024394467391962757095456, −1.17196216468015958415060738413, 0.827359178766010126476775846800, 2.70049306824499353206503091239, 3.84521141686350077366486163391, 4.65449441578218338984636778377, 5.78714416618962721228638407258, 6.21980649651919555737487062120, 7.50723949178697147582692490913, 8.499927799963496989712965944462, 9.103630164345836286948954086177, 10.12955056745516294908273055092

Graph of the ZZ-function along the critical line