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Properties

Label 2-4800-5.4-c1-0-30
Degree 22
Conductor 48004800
Sign 0.894−0.447i0.894 - 0.447i
Analytic cond. 38.328138.3281
Root an. cond. 6.190976.19097
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  + i·3-s − 4i·7-s − 9-s + 4·11-s + 2i·13-s + 6i·17-s + 4·19-s + 4·21-s − i·27-s + 2·29-s − 4·31-s + 4i·33-s − 2i·37-s − 2·39-s + 2·41-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.51i·7-s − 0.333·9-s + 1.20·11-s + 0.554i·13-s + 1.45i·17-s + 0.917·19-s + 0.872·21-s − 0.192i·27-s + 0.371·29-s − 0.718·31-s + 0.696i·33-s − 0.328i·37-s − 0.320·39-s + 0.312·41-s + ⋯

Functional equation

Λ(s)=(4800s/2 ΓC(s) L(s)=((0.894−0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4800s/2 ΓC(s+1/2) L(s)=((0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 48004800    =    26⋅3⋅522^{6} \cdot 3 \cdot 5^{2}
Sign: 0.894−0.447i0.894 - 0.447i
Analytic conductor: 38.328138.3281
Root analytic conductor: 6.190976.19097
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4800(3649,⋅)\chi_{4800} (3649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4800, ( :1/2), 0.894−0.447i)(2,\ 4800,\ (\ :1/2),\ 0.894 - 0.447i)

Particular Values

L(1)L(1) ≈\approx 2.0954999502.095499950
L(12)L(\frac12) ≈\approx 2.0954999502.095499950
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
3 1−iT 1 - iT
5 1 1
good7 1+4iT−7T2 1 + 4iT - 7T^{2}
11 1−4T+11T2 1 - 4T + 11T^{2}
13 1−2iT−13T2 1 - 2iT - 13T^{2}
17 1−6iT−17T2 1 - 6iT - 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1−2T+29T2 1 - 2T + 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1−8iT−47T2 1 - 8iT - 47T^{2}
53 1+10iT−53T2 1 + 10iT - 53T^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1+6T+61T2 1 + 6T + 61T^{2}
67 1+4iT−67T2 1 + 4iT - 67T^{2}
71 1−16T+71T2 1 - 16T + 71T^{2}
73 1+6iT−73T2 1 + 6iT - 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−12iT−83T2 1 - 12iT - 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1−14iT−97T2 1 - 14iT - 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

−8.265595995528917876287035575464, −7.66412074848188367680820773233, −6.77086699595135258450263919442, −6.34781079241062403551111301118, −5.31547676740227850415982513295, −4.35083624161163919962925768335, −3.91864028512330890369043211768, −3.31409451590162164628670898524, −1.80690725447072681773555992056, −0.907186975100964522292272236971, 0.75643389249367705915271138896, 1.88897109439793894212810085101, 2.75605359372617745715172798664, 3.44153367426976733450502456089, 4.68658356235272457345085740745, 5.49678042710000296018544707107, 5.93590539549645645863818444537, 6.88369460417687060647464891100, 7.37784477894771144501749447914, 8.330488391983104161778164168589

Graph of the ZZ-function along the critical line