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Properties

Label 2-10e3-25.19-c1-0-12
Degree 22
Conductor 10001000
Sign −0.119+0.992i-0.119 + 0.992i
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  + (−1.76 − 2.42i)3-s + 4.21i·7-s + (−1.84 + 5.68i)9-s + (−0.910 − 2.80i)11-s + (3.05 + 0.992i)13-s + (2.94 − 4.05i)17-s + (−3.38 − 2.45i)19-s + (10.2 − 7.42i)21-s + (2.02 − 0.658i)23-s + (8.48 − 2.75i)27-s + (5.74 − 4.17i)29-s + (0.918 + 0.667i)31-s + (−5.19 + 7.14i)33-s + (1.94 + 0.630i)37-s + (−2.97 − 9.15i)39-s + ⋯
L(s)  = 1  + (−1.01 − 1.39i)3-s + 1.59i·7-s + (−0.615 + 1.89i)9-s + (−0.274 − 0.845i)11-s + (0.847 + 0.275i)13-s + (0.715 − 0.984i)17-s + (−0.775 − 0.563i)19-s + (2.22 − 1.61i)21-s + (0.422 − 0.137i)23-s + (1.63 − 0.530i)27-s + (1.06 − 0.775i)29-s + (0.165 + 0.119i)31-s + (−0.903 + 1.24i)33-s + (0.319 + 0.103i)37-s + (−0.476 − 1.46i)39-s + ⋯

Functional equation

Λ(s)=(1000s/2 ΓC(s) L(s)=((−0.119+0.992i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1000s/2 ΓC(s+1/2) L(s)=((−0.119+0.992i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10001000    =    23⋅532^{3} \cdot 5^{3}
Sign: −0.119+0.992i-0.119 + 0.992i
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), −0.119+0.992i)(2,\ 1000,\ (\ :1/2),\ -0.119 + 0.992i)

Particular Values

L(1)L(1) ≈\approx 0.653625−0.736727i0.653625 - 0.736727i
L(12)L(\frac12) ≈\approx 0.653625−0.736727i0.653625 - 0.736727i
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+(1.76+2.42i)T+(−0.927+2.85i)T2 1 + (1.76 + 2.42i)T + (-0.927 + 2.85i)T^{2}
7 1−4.21iT−7T2 1 - 4.21iT - 7T^{2}
11 1+(0.910+2.80i)T+(−8.89+6.46i)T2 1 + (0.910 + 2.80i)T + (-8.89 + 6.46i)T^{2}
13 1+(−3.05−0.992i)T+(10.5+7.64i)T2 1 + (-3.05 - 0.992i)T + (10.5 + 7.64i)T^{2}
17 1+(−2.94+4.05i)T+(−5.25−16.1i)T2 1 + (-2.94 + 4.05i)T + (-5.25 - 16.1i)T^{2}
19 1+(3.38+2.45i)T+(5.87+18.0i)T2 1 + (3.38 + 2.45i)T + (5.87 + 18.0i)T^{2}
23 1+(−2.02+0.658i)T+(18.6−13.5i)T2 1 + (-2.02 + 0.658i)T + (18.6 - 13.5i)T^{2}
29 1+(−5.74+4.17i)T+(8.96−27.5i)T2 1 + (-5.74 + 4.17i)T + (8.96 - 27.5i)T^{2}
31 1+(−0.918−0.667i)T+(9.57+29.4i)T2 1 + (-0.918 - 0.667i)T + (9.57 + 29.4i)T^{2}
37 1+(−1.94−0.630i)T+(29.9+21.7i)T2 1 + (-1.94 - 0.630i)T + (29.9 + 21.7i)T^{2}
41 1+(0.323−0.996i)T+(−33.1−24.0i)T2 1 + (0.323 - 0.996i)T + (-33.1 - 24.0i)T^{2}
43 1+3.45iT−43T2 1 + 3.45iT - 43T^{2}
47 1+(3.94+5.42i)T+(−14.5+44.6i)T2 1 + (3.94 + 5.42i)T + (-14.5 + 44.6i)T^{2}
53 1+(1.86+2.56i)T+(−16.3+50.4i)T2 1 + (1.86 + 2.56i)T + (-16.3 + 50.4i)T^{2}
59 1+(−2.87+8.85i)T+(−47.7−34.6i)T2 1 + (-2.87 + 8.85i)T + (-47.7 - 34.6i)T^{2}
61 1+(−0.556−1.71i)T+(−49.3+35.8i)T2 1 + (-0.556 - 1.71i)T + (-49.3 + 35.8i)T^{2}
67 1+(−4.79+6.59i)T+(−20.7−63.7i)T2 1 + (-4.79 + 6.59i)T + (-20.7 - 63.7i)T^{2}
71 1+(−7.16+5.20i)T+(21.9−67.5i)T2 1 + (-7.16 + 5.20i)T + (21.9 - 67.5i)T^{2}
73 1+(1.00−0.327i)T+(59.0−42.9i)T2 1 + (1.00 - 0.327i)T + (59.0 - 42.9i)T^{2}
79 1+(6.21−4.51i)T+(24.4−75.1i)T2 1 + (6.21 - 4.51i)T + (24.4 - 75.1i)T^{2}
83 1+(6.11−8.42i)T+(−25.6−78.9i)T2 1 + (6.11 - 8.42i)T + (-25.6 - 78.9i)T^{2}
89 1+(−1.68−5.18i)T+(−72.0+52.3i)T2 1 + (-1.68 - 5.18i)T + (-72.0 + 52.3i)T^{2}
97 1+(9.97+13.7i)T+(−29.9+92.2i)T2 1 + (9.97 + 13.7i)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

−9.727405503884059727081664388367, −8.557402779782587096305316553727, −8.221318321813945367338547262120, −6.97183778129332800511215601875, −6.28236511250522992479402766445, −5.66958875824618303275071744905, −4.92224216039856574654021676013, −3.00379003283801769505904423689, −2.03763451820847766597908120056, −0.62823669949900980924825693629, 1.12232292002471409347888857408, 3.40471948271598021398776664271, 4.15244847715101443368013386265, 4.75410473325003961959095547543, 5.83523077244001221684329259340, 6.60717740289816626607452249335, 7.66119467798209860787249853409, 8.658518767308475384137107126045, 9.863434139315543672783154506677, 10.29166170412923682958443324334

Graph of the ZZ-function along the critical line