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Properties

Label 2-4000-8.5-c1-0-50
Degree 22
Conductor 40004000
Sign 0.914−0.404i0.914 - 0.404i
Analytic cond. 31.940131.9401
Root an. cond. 5.651565.65156
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  − 0.207i·3-s + 2.73·7-s + 2.95·9-s + 0.871i·11-s + 1.98i·13-s − 2.93·17-s − 0.905i·19-s − 0.566i·21-s + 6.50·23-s − 1.23i·27-s + 7.71i·29-s + 4.14·31-s + 0.180·33-s − 0.436i·37-s + 0.410·39-s + ⋯
L(s)  = 1  − 0.119i·3-s + 1.03·7-s + 0.985·9-s + 0.262i·11-s + 0.550i·13-s − 0.711·17-s − 0.207i·19-s − 0.123i·21-s + 1.35·23-s − 0.237i·27-s + 1.43i·29-s + 0.744·31-s + 0.0314·33-s − 0.0718i·37-s + 0.0658·39-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=((0.914−0.404i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s+1/2) L(s)=((0.914−0.404i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: 0.914−0.404i0.914 - 0.404i
Analytic conductor: 31.940131.9401
Root analytic conductor: 5.651565.65156
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4000(2001,⋅)\chi_{4000} (2001, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :1/2), 0.914−0.404i)(2,\ 4000,\ (\ :1/2),\ 0.914 - 0.404i)

Particular Values

L(1)L(1) ≈\approx 2.4334586592.433458659
L(12)L(\frac12) ≈\approx 2.4334586592.433458659
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+0.207iT−3T2 1 + 0.207iT - 3T^{2}
7 1−2.73T+7T2 1 - 2.73T + 7T^{2}
11 1−0.871iT−11T2 1 - 0.871iT - 11T^{2}
13 1−1.98iT−13T2 1 - 1.98iT - 13T^{2}
17 1+2.93T+17T2 1 + 2.93T + 17T^{2}
19 1+0.905iT−19T2 1 + 0.905iT - 19T^{2}
23 1−6.50T+23T2 1 - 6.50T + 23T^{2}
29 1−7.71iT−29T2 1 - 7.71iT - 29T^{2}
31 1−4.14T+31T2 1 - 4.14T + 31T^{2}
37 1+0.436iT−37T2 1 + 0.436iT - 37T^{2}
41 1−5.84T+41T2 1 - 5.84T + 41T^{2}
43 1+3.55iT−43T2 1 + 3.55iT - 43T^{2}
47 1+4.69T+47T2 1 + 4.69T + 47T^{2}
53 1−9.68iT−53T2 1 - 9.68iT - 53T^{2}
59 1+12.8iT−59T2 1 + 12.8iT - 59T^{2}
61 1−12.7iT−61T2 1 - 12.7iT - 61T^{2}
67 1−10.2iT−67T2 1 - 10.2iT - 67T^{2}
71 1−12.9T+71T2 1 - 12.9T + 71T^{2}
73 1+9.76T+73T2 1 + 9.76T + 73T^{2}
79 1+3.02T+79T2 1 + 3.02T + 79T^{2}
83 1+9.88iT−83T2 1 + 9.88iT - 83T^{2}
89 1+8.04T+89T2 1 + 8.04T + 89T^{2}
97 1−12.1T+97T2 1 - 12.1T + 97T^{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

−8.593880663429921169396447574418, −7.63206079089458574386750163491, −7.08370890265030669514385489550, −6.50778521031429310881532530792, −5.34047665093713371822422330200, −4.66213177074947309171826690518, −4.16357486704042919394180793602, −2.92818234273234868835291097545, −1.87763948153467500935592714677, −1.12129887753862768418249273724, 0.831205180424044530980871050196, 1.83899039927304517593549990994, 2.84978079604928203243441224962, 3.95001150558528903784443455017, 4.66139654621753170941361734562, 5.22932332028934208563084803577, 6.26579957935672803529955770655, 6.94166953859460761455679793943, 7.87430953089406589474747637815, 8.178190511834108910949517853070

Graph of the ZZ-function along the critical line