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Properties

Label 2-448-28.19-c3-0-4
Degree 22
Conductor 448448
Sign 0.957−0.288i0.957 - 0.288i
Analytic cond. 26.432826.4328
Root an. cond. 5.141285.14128
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.65 − 8.06i)3-s + (−5.11 − 2.95i)5-s + (1.92 − 18.4i)7-s + (−29.8 + 51.6i)9-s + (−0.267 + 0.154i)11-s + 43.7i·13-s + 54.9i·15-s + (−27.0 + 15.6i)17-s + (−39.7 + 68.8i)19-s + (−157. + 70.1i)21-s + (16.5 + 9.52i)23-s + (−45.0 − 78.0i)25-s + 303.·27-s + 40.1·29-s + (42.8 + 74.2i)31-s + ⋯
L(s)  = 1  + (−0.895 − 1.55i)3-s + (−0.457 − 0.264i)5-s + (0.104 − 0.994i)7-s + (−1.10 + 1.91i)9-s + (−0.00733 + 0.00423i)11-s + 0.932i·13-s + 0.946i·15-s + (−0.386 + 0.223i)17-s + (−0.479 + 0.830i)19-s + (−1.63 + 0.729i)21-s + (0.149 + 0.0863i)23-s + (−0.360 − 0.624i)25-s + 2.16·27-s + 0.256·29-s + (0.248 + 0.430i)31-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=((0.957−0.288i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+3/2) L(s)=((0.957−0.288i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: 0.957−0.288i0.957 - 0.288i
Analytic conductor: 26.432826.4328
Root analytic conductor: 5.141285.14128
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ448(383,⋅)\chi_{448} (383, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 448, ( :3/2), 0.957−0.288i)(2,\ 448,\ (\ :3/2),\ 0.957 - 0.288i)

Particular Values

L(2)L(2) ≈\approx 0.50754534340.5075453434
L(12)L(\frac12) ≈\approx 0.50754534340.5075453434
L(52)L(\frac{5}{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
7 1+(−1.92+18.4i)T 1 + (-1.92 + 18.4i)T
good3 1+(4.65+8.06i)T+(−13.5+23.3i)T2 1 + (4.65 + 8.06i)T + (-13.5 + 23.3i)T^{2}
5 1+(5.11+2.95i)T+(62.5+108.i)T2 1 + (5.11 + 2.95i)T + (62.5 + 108. i)T^{2}
11 1+(0.267−0.154i)T+(665.5−1.15e3i)T2 1 + (0.267 - 0.154i)T + (665.5 - 1.15e3i)T^{2}
13 1−43.7iT−2.19e3T2 1 - 43.7iT - 2.19e3T^{2}
17 1+(27.0−15.6i)T+(2.45e3−4.25e3i)T2 1 + (27.0 - 15.6i)T + (2.45e3 - 4.25e3i)T^{2}
19 1+(39.7−68.8i)T+(−3.42e3−5.94e3i)T2 1 + (39.7 - 68.8i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(−16.5−9.52i)T+(6.08e3+1.05e4i)T2 1 + (-16.5 - 9.52i)T + (6.08e3 + 1.05e4i)T^{2}
29 1−40.1T+2.43e4T2 1 - 40.1T + 2.43e4T^{2}
31 1+(−42.8−74.2i)T+(−1.48e4+2.57e4i)T2 1 + (-42.8 - 74.2i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(72.4−125.i)T+(−2.53e4−4.38e4i)T2 1 + (72.4 - 125. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1−254.iT−6.89e4T2 1 - 254. iT - 6.89e4T^{2}
43 1+366.iT−7.95e4T2 1 + 366. iT - 7.95e4T^{2}
47 1+(5.12−8.87i)T+(−5.19e4−8.99e4i)T2 1 + (5.12 - 8.87i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(315.+546.i)T+(−7.44e4+1.28e5i)T2 1 + (315. + 546. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(−257.−446.i)T+(−1.02e5+1.77e5i)T2 1 + (-257. - 446. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−669.−386.i)T+(1.13e5+1.96e5i)T2 1 + (-669. - 386. i)T + (1.13e5 + 1.96e5i)T^{2}
67 1+(−450.+259.i)T+(1.50e5−2.60e5i)T2 1 + (-450. + 259. i)T + (1.50e5 - 2.60e5i)T^{2}
71 1−261.iT−3.57e5T2 1 - 261. iT - 3.57e5T^{2}
73 1+(−588.+339.i)T+(1.94e5−3.36e5i)T2 1 + (-588. + 339. i)T + (1.94e5 - 3.36e5i)T^{2}
79 1+(299.+172.i)T+(2.46e5+4.26e5i)T2 1 + (299. + 172. i)T + (2.46e5 + 4.26e5i)T^{2}
83 1−805.T+5.71e5T2 1 - 805.T + 5.71e5T^{2}
89 1+(40.7+23.5i)T+(3.52e5+6.10e5i)T2 1 + (40.7 + 23.5i)T + (3.52e5 + 6.10e5i)T^{2}
97 1−763.iT−9.12e5T2 1 - 763. iT - 9.12e5T^{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

−11.00545914416135307063992674808, −10.08475361689581260998907879523, −8.497368836973818088561414647564, −7.83507770299879752825987932075, −6.86696028946125397314318041808, −6.39245144402913579663009021840, −5.08092109632802843967417934730, −3.96120629388547398068858634959, −2.02790136213562454620582986870, −0.961022829921070248312949009224, 0.23166319796105913900465715351, 2.76145151197298689353952576819, 3.88025043670143244986744078803, 4.94480832511804792488538348236, 5.61129042277035784900869875611, 6.63031632772264630677344016990, 8.109312308197776064308359217633, 9.112540511550688701343816148719, 9.770441864644129863577642845938, 10.87412670442892012440955780504

Graph of the ZZ-function along the critical line