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Properties

Label 2-48-16.13-c1-0-2
Degree 22
Conductor 4848
Sign 0.773+0.633i0.773 + 0.633i
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  + (0.635 − 1.26i)2-s + (0.707 + 0.707i)3-s + (−1.19 − 1.60i)4-s + (−2.68 + 2.68i)5-s + (1.34 − 0.443i)6-s − 2.15i·7-s + (−2.78 + 0.484i)8-s + 1.00i·9-s + (1.68 + 5.09i)10-s + (1.79 − 1.79i)11-s + (0.292 − 1.97i)12-s + (1.38 + 1.38i)13-s + (−2.72 − 1.37i)14-s − 3.79·15-s + (−1.15 + 3.82i)16-s − 0.224·17-s + ⋯
L(s)  = 1  + (0.449 − 0.893i)2-s + (0.408 + 0.408i)3-s + (−0.595 − 0.803i)4-s + (−1.20 + 1.20i)5-s + (0.548 − 0.181i)6-s − 0.816i·7-s + (−0.985 + 0.171i)8-s + 0.333i·9-s + (0.533 + 1.61i)10-s + (0.542 − 0.542i)11-s + (0.0845 − 0.571i)12-s + (0.383 + 0.383i)13-s + (−0.728 − 0.366i)14-s − 0.980·15-s + (−0.289 + 0.957i)16-s − 0.0545·17-s + ⋯

Functional equation

Λ(s)=(48s/2 ΓC(s) L(s)=((0.773+0.633i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(48s/2 ΓC(s+1/2) L(s)=((0.773+0.633i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4848    =    24⋅32^{4} \cdot 3
Sign: 0.773+0.633i0.773 + 0.633i
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), 0.773+0.633i)(2,\ 48,\ (\ :1/2),\ 0.773 + 0.633i)

Particular Values

L(1)L(1) ≈\approx 0.871540−0.311560i0.871540 - 0.311560i
L(12)L(\frac12) ≈\approx 0.871540−0.311560i0.871540 - 0.311560i
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+(−0.635+1.26i)T 1 + (-0.635 + 1.26i)T
3 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good5 1+(2.68−2.68i)T−5iT2 1 + (2.68 - 2.68i)T - 5iT^{2}
7 1+2.15iT−7T2 1 + 2.15iT - 7T^{2}
11 1+(−1.79+1.79i)T−11iT2 1 + (-1.79 + 1.79i)T - 11iT^{2}
13 1+(−1.38−1.38i)T+13iT2 1 + (-1.38 - 1.38i)T + 13iT^{2}
17 1+0.224T+17T2 1 + 0.224T + 17T^{2}
19 1+(−0.158−0.158i)T+19iT2 1 + (-0.158 - 0.158i)T + 19iT^{2}
23 1+2.82iT−23T2 1 + 2.82iT - 23T^{2}
29 1+(1.85+1.85i)T+29iT2 1 + (1.85 + 1.85i)T + 29iT^{2}
31 1−1.84T+31T2 1 - 1.84T + 31T^{2}
37 1+(3.66−3.66i)T−37iT2 1 + (3.66 - 3.66i)T - 37iT^{2}
41 1−5.88iT−41T2 1 - 5.88iT - 41T^{2}
43 1+(7.75−7.75i)T−43iT2 1 + (7.75 - 7.75i)T - 43iT^{2}
47 1+2.82T+47T2 1 + 2.82T + 47T^{2}
53 1+(−7.51+7.51i)T−53iT2 1 + (-7.51 + 7.51i)T - 53iT^{2}
59 1+(−4+4i)T−59iT2 1 + (-4 + 4i)T - 59iT^{2}
61 1+(−5.98−5.98i)T+61iT2 1 + (-5.98 - 5.98i)T + 61iT^{2}
67 1+(10.4+10.4i)T+67iT2 1 + (10.4 + 10.4i)T + 67iT^{2}
71 1−4.31iT−71T2 1 - 4.31iT - 71T^{2}
73 1−5.97iT−73T2 1 - 5.97iT - 73T^{2}
79 1−15.0T+79T2 1 - 15.0T + 79T^{2}
83 1+(10.1+10.1i)T+83iT2 1 + (10.1 + 10.1i)T + 83iT^{2}
89 1−1.42iT−89T2 1 - 1.42iT - 89T^{2}
97 1+16.3T+97T2 1 + 16.3T + 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

−15.15685271547203927047634424941, −14.41983232478527245787039181785, −13.45950518284363879693775285106, −11.73266118695565942172873152568, −11.02654065873239224617703655017, −10.01882682873606116854011464606, −8.340574390110295203834746480786, −6.67926877056311355946931200854, −4.23338644021661599711126263758, −3.26001583815327479995642235752, 3.86179068547023561844984227152, 5.37842133309248723250807203915, 7.20809824324198859285104900790, 8.381293715168107984791390773270, 9.086993693289973544211052872954, 11.88643337552233925390382200980, 12.43031386523642399712733001277, 13.52401554393183182785110398502, 15.02615018984691346279224474452, 15.60478899011562921687979644080

Graph of the ZZ-function along the critical line