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Properties

Label 2-459-153.106-c1-0-6
Degree 22
Conductor 459459
Sign 0.929+0.369i0.929 + 0.369i
Analytic cond. 3.665133.66513
Root an. cond. 1.914451.91445
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  + (−0.218 − 0.126i)2-s + (−0.968 − 1.67i)4-s + (3.83 + 1.02i)5-s + (2.05 − 0.551i)7-s + 0.995i·8-s + (−0.709 − 0.709i)10-s + (−1.14 + 0.307i)11-s + (3.09 + 5.36i)13-s + (−0.520 − 0.139i)14-s + (−1.81 + 3.13i)16-s + (−2.45 − 3.31i)17-s − 5.66i·19-s + (−1.98 − 7.41i)20-s + (0.290 + 0.0778i)22-s + (0.188 − 0.703i)23-s + ⋯
L(s)  = 1  + (−0.154 − 0.0893i)2-s + (−0.484 − 0.838i)4-s + (1.71 + 0.459i)5-s + (0.778 − 0.208i)7-s + 0.351i·8-s + (−0.224 − 0.224i)10-s + (−0.346 + 0.0928i)11-s + (0.858 + 1.48i)13-s + (−0.139 − 0.0372i)14-s + (−0.452 + 0.783i)16-s + (−0.595 − 0.803i)17-s − 1.29i·19-s + (−0.444 − 1.65i)20-s + (0.0619 + 0.0165i)22-s + (0.0392 − 0.146i)23-s + ⋯

Functional equation

Λ(s)=(459s/2 ΓC(s) L(s)=((0.929+0.369i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(459s/2 ΓC(s+1/2) L(s)=((0.929+0.369i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 459459    =    33⋅173^{3} \cdot 17
Sign: 0.929+0.369i0.929 + 0.369i
Analytic conductor: 3.665133.66513
Root analytic conductor: 1.914451.91445
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ459(208,⋅)\chi_{459} (208, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 459, ( :1/2), 0.929+0.369i)(2,\ 459,\ (\ :1/2),\ 0.929 + 0.369i)

Particular Values

L(1)L(1) ≈\approx 1.62156−0.310995i1.62156 - 0.310995i
L(12)L(\frac12) ≈\approx 1.62156−0.310995i1.62156 - 0.310995i
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)
bad3 1 1
17 1+(2.45+3.31i)T 1 + (2.45 + 3.31i)T
good2 1+(0.218+0.126i)T+(1+1.73i)T2 1 + (0.218 + 0.126i)T + (1 + 1.73i)T^{2}
5 1+(−3.83−1.02i)T+(4.33+2.5i)T2 1 + (-3.83 - 1.02i)T + (4.33 + 2.5i)T^{2}
7 1+(−2.05+0.551i)T+(6.06−3.5i)T2 1 + (-2.05 + 0.551i)T + (6.06 - 3.5i)T^{2}
11 1+(1.14−0.307i)T+(9.52−5.5i)T2 1 + (1.14 - 0.307i)T + (9.52 - 5.5i)T^{2}
13 1+(−3.09−5.36i)T+(−6.5+11.2i)T2 1 + (-3.09 - 5.36i)T + (-6.5 + 11.2i)T^{2}
19 1+5.66iT−19T2 1 + 5.66iT - 19T^{2}
23 1+(−0.188+0.703i)T+(−19.9−11.5i)T2 1 + (-0.188 + 0.703i)T + (-19.9 - 11.5i)T^{2}
29 1+(−0.799−2.98i)T+(−25.1+14.5i)T2 1 + (-0.799 - 2.98i)T + (-25.1 + 14.5i)T^{2}
31 1+(−0.0434−0.0116i)T+(26.8+15.5i)T2 1 + (-0.0434 - 0.0116i)T + (26.8 + 15.5i)T^{2}
37 1+(−1.68+1.68i)T−37iT2 1 + (-1.68 + 1.68i)T - 37iT^{2}
41 1+(−0.174+0.650i)T+(−35.5−20.5i)T2 1 + (-0.174 + 0.650i)T + (-35.5 - 20.5i)T^{2}
43 1+(0.442+0.255i)T+(21.5+37.2i)T2 1 + (0.442 + 0.255i)T + (21.5 + 37.2i)T^{2}
47 1+(−4.74+8.21i)T+(−23.5−40.7i)T2 1 + (-4.74 + 8.21i)T + (-23.5 - 40.7i)T^{2}
53 1+7.10iT−53T2 1 + 7.10iT - 53T^{2}
59 1+(7.91−4.56i)T+(29.5−51.0i)T2 1 + (7.91 - 4.56i)T + (29.5 - 51.0i)T^{2}
61 1+(5.59−1.49i)T+(52.8−30.5i)T2 1 + (5.59 - 1.49i)T + (52.8 - 30.5i)T^{2}
67 1+(1.68+2.92i)T+(−33.5+58.0i)T2 1 + (1.68 + 2.92i)T + (-33.5 + 58.0i)T^{2}
71 1+(2.32−2.32i)T−71iT2 1 + (2.32 - 2.32i)T - 71iT^{2}
73 1+(8.24−8.24i)T−73iT2 1 + (8.24 - 8.24i)T - 73iT^{2}
79 1+(0.124−0.0332i)T+(68.4−39.5i)T2 1 + (0.124 - 0.0332i)T + (68.4 - 39.5i)T^{2}
83 1+(5.37+3.10i)T+(41.5+71.8i)T2 1 + (5.37 + 3.10i)T + (41.5 + 71.8i)T^{2}
89 1+8.09T+89T2 1 + 8.09T + 89T^{2}
97 1+(−1.05−3.94i)T+(−84.0+48.5i)T2 1 + (-1.05 - 3.94i)T + (-84.0 + 48.5i)T^{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

−10.89187803746652319884107200636, −10.10979252240225599494109411965, −9.165210863860440807267835926471, −8.827718459734963039858623891937, −7.03690286083764834488021018702, −6.30728772968983756598432663810, −5.28779307087957943832397256169, −4.51083958568445782618484724248, −2.42712187370288480166860743304, −1.46834122641862635276485038944, 1.52047536682668799647628589153, 2.94785054804758133160795683632, 4.43043738345262094201423221727, 5.55916425024461958268148159801, 6.16356750013453104279419128950, 7.84948267549001615392883385137, 8.373883994935182036573603137016, 9.219087528855299942006236687120, 10.15614619939716211817580812035, 10.88197611327407949004402447276

Graph of the ZZ-function along the critical line