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Properties

Label 2-5040-5.4-c1-0-5
Degree 22
Conductor 50405040
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 40.244640.2446
Root an. cond. 6.343866.34386
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  + (−2 + i)5-s − i·7-s − 11-s − i·13-s − 3i·17-s − 4·19-s + 2i·23-s + (3 − 4i)25-s − 29-s + 6·31-s + (1 + 2i)35-s − 2i·37-s + 10·41-s − 9i·47-s − 49-s + ⋯
L(s)  = 1  + (−0.894 + 0.447i)5-s − 0.377i·7-s − 0.301·11-s − 0.277i·13-s − 0.727i·17-s − 0.917·19-s + 0.417i·23-s + (0.600 − 0.800i)25-s − 0.185·29-s + 1.07·31-s + (0.169 + 0.338i)35-s − 0.328i·37-s + 1.56·41-s − 1.31i·47-s − 0.142·49-s + ⋯

Functional equation

Λ(s)=(5040s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(5040s/2 ΓC(s+1/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 50405040    =    24⋅32⋅5⋅72^{4} \cdot 3^{2} \cdot 5 \cdot 7
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 40.244640.2446
Root analytic conductor: 6.343866.34386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ5040(1009,⋅)\chi_{5040} (1009, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 5040, ( :1/2), −0.447−0.894i)(2,\ 5040,\ (\ :1/2),\ -0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.61618882410.6161888241
L(12)L(\frac12) ≈\approx 0.61618882410.6161888241
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 1
5 1+(2−i)T 1 + (2 - i)T
7 1+iT 1 + iT
good11 1+T+11T2 1 + T + 11T^{2}
13 1+iT−13T2 1 + iT - 13T^{2}
17 1+3iT−17T2 1 + 3iT - 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1−2iT−23T2 1 - 2iT - 23T^{2}
29 1+T+29T2 1 + T + 29T^{2}
31 1−6T+31T2 1 - 6T + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1−10T+41T2 1 - 10T + 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1+9iT−47T2 1 + 9iT - 47T^{2}
53 1−14iT−53T2 1 - 14iT - 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1+4T+61T2 1 + 4T + 61T^{2}
67 1−10iT−67T2 1 - 10iT - 67T^{2}
71 1+16T+71T2 1 + 16T + 71T^{2}
73 1−10iT−73T2 1 - 10iT - 73T^{2}
79 1+11T+79T2 1 + 11T + 79T^{2}
83 1−4iT−83T2 1 - 4iT - 83T^{2}
89 1−12T+89T2 1 - 12T + 89T^{2}
97 1−19iT−97T2 1 - 19iT - 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

−8.395565788404518260586571605794, −7.57460813863053973245246217736, −7.26736350244204974636210448635, −6.37822251676539880714682177670, −5.62036477641083055526104219028, −4.56039797867431465394370343944, −4.10445279714158812480481183592, −3.10962613931106203746762076164, −2.44102233786440693658150136763, −0.965779983167552295787105672169, 0.19847007308049571313134729690, 1.52016929022024017850096624061, 2.60085696512579932956849030651, 3.51293061900044232850777348233, 4.42920133830405890444571571155, 4.80980032008825824645681724315, 5.97458484732942600484325972926, 6.43984449003284318264038877093, 7.51065617365675916708942451567, 7.982300696960332248178305040233

Graph of the ZZ-function along the critical line