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Properties

Label 2-48-16.13-c1-0-1
Degree 22
Conductor 4848
Sign 0.762−0.646i0.762 - 0.646i
Analytic cond. 0.3832810.383281
Root an. cond. 0.6190970.619097
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  + (0.874 + 1.11i)2-s + (−0.707 − 0.707i)3-s + (−0.470 + 1.94i)4-s + (−0.334 + 0.334i)5-s + (0.167 − 1.40i)6-s − 4.55i·7-s + (−2.57 + 1.17i)8-s + 1.00i·9-s + (−0.665 − 0.0793i)10-s + (−2.47 + 2.47i)11-s + (1.70 − 1.04i)12-s + (−0.0594 − 0.0594i)13-s + (5.06 − 3.98i)14-s + 0.473·15-s + (−3.55 − 1.82i)16-s + 3.61·17-s + ⋯
L(s)  = 1  + (0.618 + 0.785i)2-s + (−0.408 − 0.408i)3-s + (−0.235 + 0.971i)4-s + (−0.149 + 0.149i)5-s + (0.0683 − 0.573i)6-s − 1.72i·7-s + (−0.909 + 0.416i)8-s + 0.333i·9-s + (−0.210 − 0.0250i)10-s + (−0.745 + 0.745i)11-s + (0.492 − 0.300i)12-s + (−0.0164 − 0.0164i)13-s + (1.35 − 1.06i)14-s + 0.122·15-s + (−0.889 − 0.457i)16-s + 0.877·17-s + ⋯

Functional equation

Λ(s)=(48s/2 ΓC(s) L(s)=((0.762−0.646i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(48s/2 ΓC(s+1/2) L(s)=((0.762−0.646i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4848    =    24⋅32^{4} \cdot 3
Sign: 0.762−0.646i0.762 - 0.646i
Analytic conductor: 0.3832810.383281
Root analytic conductor: 0.6190970.619097
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ48(13,⋅)\chi_{48} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 48, ( :1/2), 0.762−0.646i)(2,\ 48,\ (\ :1/2),\ 0.762 - 0.646i)

Particular Values

L(1)L(1) ≈\approx 0.859126+0.315241i0.859126 + 0.315241i
L(12)L(\frac12) ≈\approx 0.859126+0.315241i0.859126 + 0.315241i
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+(−0.874−1.11i)T 1 + (-0.874 - 1.11i)T
3 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good5 1+(0.334−0.334i)T−5iT2 1 + (0.334 - 0.334i)T - 5iT^{2}
7 1+4.55iT−7T2 1 + 4.55iT - 7T^{2}
11 1+(2.47−2.47i)T−11iT2 1 + (2.47 - 2.47i)T - 11iT^{2}
13 1+(0.0594+0.0594i)T+13iT2 1 + (0.0594 + 0.0594i)T + 13iT^{2}
17 1−3.61T+17T2 1 - 3.61T + 17T^{2}
19 1+(−2.55−2.55i)T+19iT2 1 + (-2.55 - 2.55i)T + 19iT^{2}
23 1−2.82iT−23T2 1 - 2.82iT - 23T^{2}
29 1+(5.16+5.16i)T+29iT2 1 + (5.16 + 5.16i)T + 29iT^{2}
31 1+0.557T+31T2 1 + 0.557T + 31T^{2}
37 1+(−4.38+4.38i)T−37iT2 1 + (-4.38 + 4.38i)T - 37iT^{2}
41 1+9.27iT−41T2 1 + 9.27iT - 41T^{2}
43 1+(1.61−1.61i)T−43iT2 1 + (1.61 - 1.61i)T - 43iT^{2}
47 1−2.82T+47T2 1 - 2.82T + 47T^{2}
53 1+(0.493−0.493i)T−53iT2 1 + (0.493 - 0.493i)T - 53iT^{2}
59 1+(−4+4i)T−59iT2 1 + (-4 + 4i)T - 59iT^{2}
61 1+(−2.72−2.72i)T+61iT2 1 + (-2.72 - 2.72i)T + 61iT^{2}
67 1+(−3.77−3.77i)T+67iT2 1 + (-3.77 - 3.77i)T + 67iT^{2}
71 1−9.11iT−71T2 1 - 9.11iT - 71T^{2}
73 1+0.541iT−73T2 1 + 0.541iT - 73T^{2}
79 1+10.9T+79T2 1 + 10.9T + 79T^{2}
83 1+(10.6+10.6i)T+83iT2 1 + (10.6 + 10.6i)T + 83iT^{2}
89 1−14.6iT−89T2 1 - 14.6iT - 89T^{2}
97 1−4.31T+97T2 1 - 4.31T + 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

−15.89760953308572664773995942205, −14.57766008330093553152231520766, −13.57402103460834438797867056377, −12.73589645070448832286116986788, −11.36083120799465652053750283849, −9.954645857861525156894588031065, −7.66130997167112872618999970533, −7.25469706611507216003136633047, −5.50016234533184009125213007544, −3.88431800021340996070071180644, 2.93277313035644301932611142198, 5.04018784539916012716801928425, 5.97148374437644136007961728477, 8.596318950109452418931733201385, 9.800984711243321477277656143961, 11.16531769844445068984838941454, 12.03702935072545998173445842602, 12.97197468381363123603374713277, 14.48970710928985023142069963813, 15.45259046146594049541990349980

Graph of the ZZ-function along the critical line