Properties

Label 2-2e2-4.3-c44-0-19
Degree 22
Conductor 44
Sign 0.2090.977i-0.209 - 0.977i
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  + (2.63e6 − 3.26e6i)2-s + 5.11e9i·3-s + (−3.68e12 − 1.72e13i)4-s − 1.83e15·5-s + (1.66e16 + 1.34e16i)6-s − 1.24e18i·7-s + (−6.58e19 − 3.33e19i)8-s + 9.58e20·9-s + (−4.84e21 + 5.98e21i)10-s − 1.41e23i·11-s + (8.79e22 − 1.88e22i)12-s − 3.73e24·13-s + (−4.06e24 − 3.28e24i)14-s − 9.38e24i·15-s + (−2.82e26 + 1.26e26i)16-s + 9.62e26·17-s + ⋯
L(s)  = 1  + (0.628 − 0.777i)2-s + 0.162i·3-s + (−0.209 − 0.977i)4-s − 0.770·5-s + (0.126 + 0.102i)6-s − 0.318i·7-s + (−0.892 − 0.451i)8-s + 0.973·9-s + (−0.484 + 0.598i)10-s − 1.73i·11-s + (0.159 − 0.0341i)12-s − 1.16·13-s + (−0.247 − 0.200i)14-s − 0.125i·15-s + (−0.912 + 0.409i)16-s + 0.819·17-s + ⋯

Functional equation

Λ(s)=(4s/2ΓC(s)L(s)=((0.2090.977i)Λ(45s)\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(45-s) \end{aligned}
Λ(s)=(4s/2ΓC(s+22)L(s)=((0.2090.977i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+22) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 44    =    222^{2}
Sign: 0.2090.977i-0.209 - 0.977i
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), 0.2090.977i)(2,\ 4,\ (\ :22),\ -0.209 - 0.977i)

Particular Values

L(452)L(\frac{45}{2}) \approx 0.31502987060.3150298706
L(12)L(\frac12) \approx 0.31502987060.3150298706
L(23)L(23) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(2.63e6+3.26e6i)T 1 + (-2.63e6 + 3.26e6i)T
good3 15.11e9iT9.84e20T2 1 - 5.11e9iT - 9.84e20T^{2}
5 1+1.83e15T+5.68e30T2 1 + 1.83e15T + 5.68e30T^{2}
7 1+1.24e18iT1.52e37T2 1 + 1.24e18iT - 1.52e37T^{2}
11 1+1.41e23iT6.62e45T2 1 + 1.41e23iT - 6.62e45T^{2}
13 1+3.73e24T+1.03e49T2 1 + 3.73e24T + 1.03e49T^{2}
17 19.62e26T+1.37e54T2 1 - 9.62e26T + 1.37e54T^{2}
19 17.45e27iT1.84e56T2 1 - 7.45e27iT - 1.84e56T^{2}
23 17.93e29iT8.24e59T2 1 - 7.93e29iT - 8.24e59T^{2}
29 1+2.33e32T+2.21e64T2 1 + 2.33e32T + 2.21e64T^{2}
31 14.48e31iT4.16e65T2 1 - 4.48e31iT - 4.16e65T^{2}
37 11.89e34T+1.00e69T2 1 - 1.89e34T + 1.00e69T^{2}
41 1+1.40e35T+9.17e70T2 1 + 1.40e35T + 9.17e70T^{2}
43 14.14e35iT7.45e71T2 1 - 4.14e35iT - 7.45e71T^{2}
47 16.22e36iT3.73e73T2 1 - 6.22e36iT - 3.73e73T^{2}
53 1+5.66e37T+7.38e75T2 1 + 5.66e37T + 7.38e75T^{2}
59 11.58e39iT8.26e77T2 1 - 1.58e39iT - 8.26e77T^{2}
61 1+1.71e39T+3.58e78T2 1 + 1.71e39T + 3.58e78T^{2}
67 1+1.52e39iT2.22e80T2 1 + 1.52e39iT - 2.22e80T^{2}
71 1+5.26e40iT2.85e81T2 1 + 5.26e40iT - 2.85e81T^{2}
73 11.04e40T+9.68e81T2 1 - 1.04e40T + 9.68e81T^{2}
79 1+9.56e41iT3.13e83T2 1 + 9.56e41iT - 3.13e83T^{2}
83 12.60e42iT2.75e84T2 1 - 2.60e42iT - 2.75e84T^{2}
89 1+8.08e42T+5.93e85T2 1 + 8.08e42T + 5.93e85T^{2}
97 11.49e43T+2.61e87T2 1 - 1.49e43T + 2.61e87T^{2}
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   L(s)=p j=12(1αj,pps)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

−13.59963757794294174449701679085, −12.17918113185564577444764938711, −10.96634742737830741467507870658, −9.607627013765662916220607478734, −7.60762907406117819143479054566, −5.67268923073270960556232288887, −4.15433897659077121845474108859, −3.19767319663320608582352157741, −1.34338978916771176861350199351, −0.06861134077647163687549871883, 2.21412528086000797226096915695, 4.00915447488166447720199212903, 5.02051359277398581135902019997, 6.99118380974996308471205876936, 7.71071505849247338464671616416, 9.659059536732959385500987105281, 12.03395336168938900183308723869, 12.79686170870166627770982776401, 14.76435124379400235857499451726, 15.53386248257781804060050583490

Graph of the ZZ-function along the critical line