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Properties

Label 2-3744-104.77-c1-0-6
Degree 22
Conductor 37443744
Sign −0.980−0.196i-0.980 - 0.196i
Analytic cond. 29.895929.8959
Root an. cond. 5.467725.46772
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  + 5-s + 3i·7-s − 2·11-s + (3 − 2i)13-s − 3·17-s − 6·23-s − 4·25-s + 6i·29-s + 3i·35-s − 3·37-s − 10i·41-s + 9i·43-s + 7i·47-s − 2·49-s − 6i·53-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.13i·7-s − 0.603·11-s + (0.832 − 0.554i)13-s − 0.727·17-s − 1.25·23-s − 0.800·25-s + 1.11i·29-s + 0.507i·35-s − 0.493·37-s − 1.56i·41-s + 1.37i·43-s + 1.02i·47-s − 0.285·49-s − 0.824i·53-s + ⋯

Functional equation

Λ(s)=(3744s/2 ΓC(s) L(s)=((−0.980−0.196i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3744s/2 ΓC(s+1/2) L(s)=((−0.980−0.196i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37443744    =    25⋅32⋅132^{5} \cdot 3^{2} \cdot 13
Sign: −0.980−0.196i-0.980 - 0.196i
Analytic conductor: 29.895929.8959
Root analytic conductor: 5.467725.46772
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3744(1585,⋅)\chi_{3744} (1585, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3744, ( :1/2), −0.980−0.196i)(2,\ 3744,\ (\ :1/2),\ -0.980 - 0.196i)

Particular Values

L(1)L(1) ≈\approx 0.54842682750.5484268275
L(12)L(\frac12) ≈\approx 0.54842682750.5484268275
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
13 1+(−3+2i)T 1 + (-3 + 2i)T
good5 1−T+5T2 1 - T + 5T^{2}
7 1−3iT−7T2 1 - 3iT - 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
17 1+3T+17T2 1 + 3T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
29 1−6iT−29T2 1 - 6iT - 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1+3T+37T2 1 + 3T + 37T^{2}
41 1+10iT−41T2 1 + 10iT - 41T^{2}
43 1−9iT−43T2 1 - 9iT - 43T^{2}
47 1−7iT−47T2 1 - 7iT - 47T^{2}
53 1+6iT−53T2 1 + 6iT - 53T^{2}
59 1+10T+59T2 1 + 10T + 59T^{2}
61 1−10iT−61T2 1 - 10iT - 61T^{2}
67 1+12T+67T2 1 + 12T + 67T^{2}
71 1−5iT−71T2 1 - 5iT - 71T^{2}
73 1+6iT−73T2 1 + 6iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−16T+83T2 1 - 16T + 83T^{2}
89 1−4iT−89T2 1 - 4iT - 89T^{2}
97 1+18iT−97T2 1 + 18iT - 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

−8.869512354462601399654551415056, −8.211010742775666279718371238507, −7.50517299867787095471162236233, −6.41158513193436131253351348246, −5.84981594954039124744514809038, −5.32218129745857483059681628483, −4.32113556113918426629979024348, −3.27840343935163274625854080462, −2.42769242749467014929904873173, −1.59996790756401227109921359506, 0.14748134850072264780085898210, 1.53107726336924127940583740583, 2.40029102597249136560902495368, 3.69055430505702448338018136457, 4.19373498968219773073118436251, 5.11433187728673941286343198260, 6.14448099354391109716302156388, 6.52346341183531242304633628487, 7.57128160868109870392226098742, 8.013296097028327413479950037862

Graph of the ZZ-function along the critical line