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Properties

Label 2-3744-8.5-c1-0-29
Degree 22
Conductor 37443744
Sign 0.707+0.707i0.707 + 0.707i
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  − 3i·5-s − 3·7-s + i·13-s + 7·17-s + 4i·19-s + 4·23-s − 4·25-s + 4i·29-s + 8·31-s + 9i·35-s + 7i·37-s − 2·41-s + i·43-s − 7·47-s + 2·49-s + ⋯
L(s)  = 1  − 1.34i·5-s − 1.13·7-s + 0.277i·13-s + 1.69·17-s + 0.917i·19-s + 0.834·23-s − 0.800·25-s + 0.742i·29-s + 1.43·31-s + 1.52i·35-s + 1.15i·37-s − 0.312·41-s + 0.152i·43-s − 1.02·47-s + 0.285·49-s + ⋯

Functional equation

Λ(s)=(3744s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3744s/2 ΓC(s+1/2) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37443744    =    25⋅32⋅132^{5} \cdot 3^{2} \cdot 13
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 29.895929.8959
Root analytic conductor: 5.467725.46772
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3744(1873,⋅)\chi_{3744} (1873, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3744, ( :1/2), 0.707+0.707i)(2,\ 3744,\ (\ :1/2),\ 0.707 + 0.707i)

Particular Values

L(1)L(1) ≈\approx 1.6620866791.662086679
L(12)L(\frac12) ≈\approx 1.6620866791.662086679
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−iT 1 - iT
good5 1+3iT−5T2 1 + 3iT - 5T^{2}
7 1+3T+7T2 1 + 3T + 7T^{2}
11 1−11T2 1 - 11T^{2}
17 1−7T+17T2 1 - 7T + 17T^{2}
19 1−4iT−19T2 1 - 4iT - 19T^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
29 1−4iT−29T2 1 - 4iT - 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1−7iT−37T2 1 - 7iT - 37T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1−iT−43T2 1 - iT - 43T^{2}
47 1+7T+47T2 1 + 7T + 47T^{2}
53 1+4iT−53T2 1 + 4iT - 53T^{2}
59 1+14iT−59T2 1 + 14iT - 59T^{2}
61 1+10iT−61T2 1 + 10iT - 61T^{2}
67 1+2iT−67T2 1 + 2iT - 67T^{2}
71 1+3T+71T2 1 + 3T + 71T^{2}
73 1−14T+73T2 1 - 14T + 73T^{2}
79 1−10T+79T2 1 - 10T + 79T^{2}
83 1−14iT−83T2 1 - 14iT - 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1−8T+97T2 1 - 8T + 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

−8.249013359938070707737154342917, −8.024390805779863929082350778109, −6.78827086121344587444916827575, −6.24822188770034146979853787686, −5.24572452337337432413576212737, −4.83755644608983131410532151537, −3.63313328700375840805774317030, −3.11027091550498953658062136111, −1.61379954473299415405957426706, −0.72156226730330465486695067466, 0.814596216461762226005109417355, 2.50667611440250328149732812498, 3.06833981906607753311022033292, 3.66409819739302987864936745574, 4.85360871593317142099655212774, 5.89642221190825359384461806898, 6.36091404248984226508671355346, 7.18313079836657899187790598726, 7.58448174844116243181029221418, 8.634062321997018886648688134846

Graph of the ZZ-function along the critical line