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Properties

Label 2-2e2-4.3-c44-0-14
Degree 22
Conductor 44
Sign 0.881+0.472i0.881 + 0.472i
Analytic cond. 49.047849.0478
Root an. cond. 7.003427.00342
Motivic weight 4444
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.06e6 + 1.02e6i)2-s + 4.78e10i·3-s + (1.55e13 + 8.31e12i)4-s − 4.02e15·5-s + (−4.88e16 + 1.94e17i)6-s − 5.21e18i·7-s + (5.45e19 + 4.96e19i)8-s − 1.30e21·9-s + (−1.63e22 − 4.10e21i)10-s − 5.34e22i·11-s + (−3.97e23 + 7.41e23i)12-s + 3.07e24·13-s + (5.32e24 − 2.12e25i)14-s − 1.92e26i·15-s + (1.71e26 + 2.57e26i)16-s − 1.03e27·17-s + ⋯
L(s)  = 1  + (0.969 + 0.243i)2-s + 1.52i·3-s + (0.881 + 0.472i)4-s − 1.68·5-s + (−0.371 + 1.47i)6-s − 1.33i·7-s + (0.739 + 0.672i)8-s − 1.32·9-s + (−1.63 − 0.410i)10-s − 0.656i·11-s + (−0.719 + 1.34i)12-s + 0.956·13-s + (0.324 − 1.29i)14-s − 2.57i·15-s + (0.553 + 0.832i)16-s − 0.885·17-s + ⋯

Functional equation

Λ(s)=(4s/2 ΓC(s) L(s)=((0.881+0.472i) Λ‾(45−s)\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(45-s) \end{aligned}
Λ(s)=(4s/2 ΓC(s+22) L(s)=((0.881+0.472i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+22) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 44    =    222^{2}
Sign: 0.881+0.472i0.881 + 0.472i
Analytic conductor: 49.047849.0478
Root analytic conductor: 7.003427.00342
Motivic weight: 4444
Rational: no
Arithmetic: yes
Character: χ4(3,⋅)\chi_{4} (3, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4, ( :22), 0.881+0.472i)(2,\ 4,\ (\ :22),\ 0.881 + 0.472i)

Particular Values

L(452)L(\frac{45}{2}) ≈\approx 1.7622837131.762283713
L(12)L(\frac12) ≈\approx 1.7622837131.762283713
L(23)L(23) 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+(−4.06e6−1.02e6i)T 1 + (-4.06e6 - 1.02e6i)T
good3 1−4.78e10iT−9.84e20T2 1 - 4.78e10iT - 9.84e20T^{2}
5 1+4.02e15T+5.68e30T2 1 + 4.02e15T + 5.68e30T^{2}
7 1+5.21e18iT−1.52e37T2 1 + 5.21e18iT - 1.52e37T^{2}
11 1+5.34e22iT−6.62e45T2 1 + 5.34e22iT - 6.62e45T^{2}
13 1−3.07e24T+1.03e49T2 1 - 3.07e24T + 1.03e49T^{2}
17 1+1.03e27T+1.37e54T2 1 + 1.03e27T + 1.37e54T^{2}
19 1+1.75e28iT−1.84e56T2 1 + 1.75e28iT - 1.84e56T^{2}
23 1+1.30e29iT−8.24e59T2 1 + 1.30e29iT - 8.24e59T^{2}
29 1+1.16e32T+2.21e64T2 1 + 1.16e32T + 2.21e64T^{2}
31 1−2.21e32iT−4.16e65T2 1 - 2.21e32iT - 4.16e65T^{2}
37 1−1.46e34T+1.00e69T2 1 - 1.46e34T + 1.00e69T^{2}
41 1+2.56e35T+9.17e70T2 1 + 2.56e35T + 9.17e70T^{2}
43 1−7.73e34iT−7.45e71T2 1 - 7.73e34iT - 7.45e71T^{2}
47 1+1.06e37iT−3.73e73T2 1 + 1.06e37iT - 3.73e73T^{2}
53 1−5.04e37T+7.38e75T2 1 - 5.04e37T + 7.38e75T^{2}
59 1+1.45e39iT−8.26e77T2 1 + 1.45e39iT - 8.26e77T^{2}
61 1−2.53e39T+3.58e78T2 1 - 2.53e39T + 3.58e78T^{2}
67 1+1.88e40iT−2.22e80T2 1 + 1.88e40iT - 2.22e80T^{2}
71 1+6.82e40iT−2.85e81T2 1 + 6.82e40iT - 2.85e81T^{2}
73 1+5.18e40T+9.68e81T2 1 + 5.18e40T + 9.68e81T^{2}
79 1+1.41e41iT−3.13e83T2 1 + 1.41e41iT - 3.13e83T^{2}
83 1−2.71e41iT−2.75e84T2 1 - 2.71e41iT - 2.75e84T^{2}
89 1+5.08e42T+5.93e85T2 1 + 5.08e42T + 5.93e85T^{2}
97 1+2.11e43T+2.61e87T2 1 + 2.11e43T + 2.61e87T^{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

−15.07170672153672844095979623335, −13.48053377120867121856834642847, −11.32411772896041422972284793730, −10.82434914633751778971238699097, −8.454726154844239531821935665692, −6.95379371595648361853236207846, −4.83068417071982694024967430554, −3.94095464020444526449317326390, −3.40289829374845905655311029416, −0.35047397668912534562144092172, 1.31410637660700281384383909691, 2.53497016082375979071352675982, 4.01335976121415520771442683049, 5.86345465688891884619685939598, 7.15335099116746050129664936446, 8.304972327016775437926869800487, 11.40881633678683976565977778410, 12.13404653906992732178133286895, 13.01465382338941740265379463455, 14.83474096729184455901036377053

Graph of the ZZ-function along the critical line