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Properties

Label 2-3744-104.77-c1-0-45
Degree 22
Conductor 37443744
Sign −0.196+0.980i-0.196 + 0.980i
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)=((−0.196+0.980i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3744s/2 ΓC(s+1/2) L(s)=((−0.196+0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37443744    =    25⋅32⋅132^{5} \cdot 3^{2} \cdot 13
Sign: −0.196+0.980i-0.196 + 0.980i
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), −0.196+0.980i)(2,\ 3744,\ (\ :1/2),\ -0.196 + 0.980i)

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

−8.350174052216300662525422634468, −7.62046902975398087474564107542, −6.82162951728595368314451217572, −6.03935617624287427887833445924, −5.39696187218225109748207916278, −4.33348688716486543184726182852, −3.82971133827768777415195765379, −2.48037220080733794914316177378, −1.97353877672884316303680034318, −0.17471963942233941914089943808, 1.05229788046133669700911800195, 2.27692366594132568685905235280, 3.42878837016600000907123002105, 4.12096496599099372486847831525, 4.74963001396955454767547963126, 5.77697655521623084653260108503, 6.69206385819191479688383533557, 7.22895213128669574766760036898, 7.971741426722753366922585104685, 8.521627997179533620272873888961

Graph of the ZZ-function along the critical line