Properties

Label 2-3744-13.12-c1-0-52
Degree 22
Conductor 37443744
Sign 0.554+0.832i-0.554 + 0.832i
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·5-s + (−3 − 2i)13-s + 2·17-s − 11·25-s − 10·29-s − 12i·37-s + 8i·41-s + 7·49-s − 14·53-s − 10·61-s + (8 − 12i)65-s + 16i·73-s + 8i·85-s − 16i·89-s − 8i·97-s + ⋯
L(s)  = 1  + 1.78i·5-s + (−0.832 − 0.554i)13-s + 0.485·17-s − 2.20·25-s − 1.85·29-s − 1.97i·37-s + 1.24i·41-s + 49-s − 1.92·53-s − 1.28·61-s + (0.992 − 1.48i)65-s + 1.87i·73-s + 0.867i·85-s − 1.69i·89-s − 0.812i·97-s + ⋯

Functional equation

Λ(s)=(3744s/2ΓC(s)L(s)=((0.554+0.832i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3744s/2ΓC(s+1/2)L(s)=((0.554+0.832i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37443744    =    2532132^{5} \cdot 3^{2} \cdot 13
Sign: 0.554+0.832i-0.554 + 0.832i
Analytic conductor: 29.895929.8959
Root analytic conductor: 5.467725.46772
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3744(3457,)\chi_{3744} (3457, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 3744, ( :1/2), 0.554+0.832i)(2,\ 3744,\ (\ :1/2),\ -0.554 + 0.832i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)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 14iT5T2 1 - 4iT - 5T^{2}
7 17T2 1 - 7T^{2}
11 111T2 1 - 11T^{2}
17 12T+17T2 1 - 2T + 17T^{2}
19 119T2 1 - 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1+10T+29T2 1 + 10T + 29T^{2}
31 131T2 1 - 31T^{2}
37 1+12iT37T2 1 + 12iT - 37T^{2}
41 18iT41T2 1 - 8iT - 41T^{2}
43 1+43T2 1 + 43T^{2}
47 147T2 1 - 47T^{2}
53 1+14T+53T2 1 + 14T + 53T^{2}
59 159T2 1 - 59T^{2}
61 1+10T+61T2 1 + 10T + 61T^{2}
67 167T2 1 - 67T^{2}
71 171T2 1 - 71T^{2}
73 116iT73T2 1 - 16iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 183T2 1 - 83T^{2}
89 1+16iT89T2 1 + 16iT - 89T^{2}
97 1+8iT97T2 1 + 8iT - 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−7.920250557861090988000720899088, −7.47349600087989102088205715604, −6.94345609829164421040776940499, −6.00867871883172260252049158466, −5.50262499783384054777244257763, −4.27532854977663050671091326032, −3.38347913481718259752106087415, −2.77348086636941653821329988722, −1.87648676244566357541774297229, 0, 1.26214535222265802680455156179, 2.09023021001683883419116091656, 3.45790316511305003191858201870, 4.36913661305886093985718531011, 4.98250105396219012433013251598, 5.56776528716580331474882766751, 6.47969997481039407164373511739, 7.57087221163524274035714087100, 7.973832741044919213494427292927

Graph of the ZZ-function along the critical line