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Properties

Label 2-48-16.13-c1-0-0
Degree 22
Conductor 4848
Sign 0.995+0.0985i0.995 + 0.0985i
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  + (−1.34 − 0.443i)2-s + (0.707 + 0.707i)3-s + (1.60 + 1.19i)4-s + (1.27 − 1.27i)5-s + (−0.635 − 1.26i)6-s + 0.158i·7-s + (−1.62 − 2.31i)8-s + 1.00i·9-s + (−2.27 + 1.14i)10-s + (−3.79 + 3.79i)11-s + (0.292 + 1.97i)12-s + (−4.21 − 4.21i)13-s + (0.0705 − 0.213i)14-s + 1.79·15-s + (1.15 + 3.82i)16-s + 3.05·17-s + ⋯
L(s)  = 1  + (−0.949 − 0.313i)2-s + (0.408 + 0.408i)3-s + (0.803 + 0.595i)4-s + (0.568 − 0.568i)5-s + (−0.259 − 0.515i)6-s + 0.0600i·7-s + (−0.575 − 0.817i)8-s + 0.333i·9-s + (−0.718 + 0.361i)10-s + (−1.14 + 1.14i)11-s + (0.0845 + 0.571i)12-s + (−1.16 − 1.16i)13-s + (0.0188 − 0.0570i)14-s + 0.464·15-s + (0.289 + 0.957i)16-s + 0.740·17-s + ⋯

Functional equation

Λ(s)=(48s/2 ΓC(s) L(s)=((0.995+0.0985i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0985i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(48s/2 ΓC(s+1/2) L(s)=((0.995+0.0985i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0985i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4848    =    24⋅32^{4} \cdot 3
Sign: 0.995+0.0985i0.995 + 0.0985i
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), 0.995+0.0985i)(2,\ 48,\ (\ :1/2),\ 0.995 + 0.0985i)

Particular Values

L(1)L(1) ≈\approx 0.645225−0.0318835i0.645225 - 0.0318835i
L(12)L(\frac12) ≈\approx 0.645225−0.0318835i0.645225 - 0.0318835i
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+(1.34+0.443i)T 1 + (1.34 + 0.443i)T
3 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good5 1+(−1.27+1.27i)T−5iT2 1 + (-1.27 + 1.27i)T - 5iT^{2}
7 1−0.158iT−7T2 1 - 0.158iT - 7T^{2}
11 1+(3.79−3.79i)T−11iT2 1 + (3.79 - 3.79i)T - 11iT^{2}
13 1+(4.21+4.21i)T+13iT2 1 + (4.21 + 4.21i)T + 13iT^{2}
17 1−3.05T+17T2 1 - 3.05T + 17T^{2}
19 1+(2.15+2.15i)T+19iT2 1 + (2.15 + 2.15i)T + 19iT^{2}
23 1+2.82iT−23T2 1 + 2.82iT - 23T^{2}
29 1+(−2.09−2.09i)T+29iT2 1 + (-2.09 - 2.09i)T + 29iT^{2}
31 1−4.15T+31T2 1 - 4.15T + 31T^{2}
37 1+(5.98−5.98i)T−37iT2 1 + (5.98 - 5.98i)T - 37iT^{2}
41 1−2.60iT−41T2 1 - 2.60iT - 41T^{2}
43 1+(−5.75+5.75i)T−43iT2 1 + (-5.75 + 5.75i)T - 43iT^{2}
47 1+2.82T+47T2 1 + 2.82T + 47T^{2}
53 1+(−3.55+3.55i)T−53iT2 1 + (-3.55 + 3.55i)T - 53iT^{2}
59 1+(−4+4i)T−59iT2 1 + (-4 + 4i)T - 59iT^{2}
61 1+(−3.66−3.66i)T+61iT2 1 + (-3.66 - 3.66i)T + 61iT^{2}
67 1+(−0.767−0.767i)T+67iT2 1 + (-0.767 - 0.767i)T + 67iT^{2}
71 1+0.317iT−71T2 1 + 0.317iT - 71T^{2}
73 1−1.33iT−73T2 1 - 1.33iT - 73T^{2}
79 1+9.69T+79T2 1 + 9.69T + 79T^{2}
83 1+(−0.115−0.115i)T+83iT2 1 + (-0.115 - 0.115i)T + 83iT^{2}
89 1+14.3iT−89T2 1 + 14.3iT - 89T^{2}
97 1+0.571T+97T2 1 + 0.571T + 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

−15.74639593513314400293128352482, −14.91568359895856198804697804233, −13.07398197365761648406200420026, −12.27148432817518999734011634912, −10.36853297439090355671021383802, −9.883707634985119969315230270295, −8.517736335457635755693965821627, −7.36084009320182875941460385472, −5.11635332520457461218234470740, −2.58012679175009916972193838344, 2.49704459496081741143324106588, 5.81495267194521250421158340232, 7.14995200691863597812186463687, 8.305522493472280612478009275388, 9.671708471960232416786633649342, 10.67469645814641764617262851608, 12.08375113252286482808334836731, 13.82099296280306575777526224297, 14.53297203486061588609015430457, 15.88964805636840472698189183880

Graph of the ZZ-function along the critical line