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Properties

Label 2-528-33.29-c1-0-5
Degree 22
Conductor 528528
Sign −0.315−0.948i-0.315 - 0.948i
Analytic cond. 4.216104.21610
Root an. cond. 2.053312.05331
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  + (0.309 + 1.70i)3-s + (0.442 + 0.609i)5-s + (0.442 − 0.143i)7-s + (−2.80 + 1.05i)9-s + (2.46 + 2.21i)11-s + (−3.12 + 4.29i)13-s + (−0.901 + 0.942i)15-s + (2.99 − 2.17i)17-s + (−2.02 − 0.659i)19-s + (0.381 + 0.710i)21-s + 6.24i·23-s + (1.36 − 4.21i)25-s + (−2.66 − 4.46i)27-s + (3.09 + 9.53i)29-s + (−2.96 − 2.15i)31-s + ⋯
L(s)  = 1  + (0.178 + 0.983i)3-s + (0.197 + 0.272i)5-s + (0.167 − 0.0543i)7-s + (−0.936 + 0.351i)9-s + (0.744 + 0.668i)11-s + (−0.865 + 1.19i)13-s + (−0.232 + 0.243i)15-s + (0.726 − 0.527i)17-s + (−0.465 − 0.151i)19-s + (0.0833 + 0.154i)21-s + 1.30i·23-s + (0.273 − 0.843i)25-s + (−0.512 − 0.858i)27-s + (0.575 + 1.77i)29-s + (−0.532 − 0.386i)31-s + ⋯

Functional equation

Λ(s)=(528s/2 ΓC(s) L(s)=((−0.315−0.948i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(528s/2 ΓC(s+1/2) L(s)=((−0.315−0.948i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 528528    =    24⋅3⋅112^{4} \cdot 3 \cdot 11
Sign: −0.315−0.948i-0.315 - 0.948i
Analytic conductor: 4.216104.21610
Root analytic conductor: 2.053312.05331
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ528(161,⋅)\chi_{528} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 528, ( :1/2), −0.315−0.948i)(2,\ 528,\ (\ :1/2),\ -0.315 - 0.948i)

Particular Values

L(1)L(1) ≈\approx 0.850430+1.17871i0.850430 + 1.17871i
L(12)L(\frac12) ≈\approx 0.850430+1.17871i0.850430 + 1.17871i
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
3 1+(−0.309−1.70i)T 1 + (-0.309 - 1.70i)T
11 1+(−2.46−2.21i)T 1 + (-2.46 - 2.21i)T
good5 1+(−0.442−0.609i)T+(−1.54+4.75i)T2 1 + (-0.442 - 0.609i)T + (-1.54 + 4.75i)T^{2}
7 1+(−0.442+0.143i)T+(5.66−4.11i)T2 1 + (-0.442 + 0.143i)T + (5.66 - 4.11i)T^{2}
13 1+(3.12−4.29i)T+(−4.01−12.3i)T2 1 + (3.12 - 4.29i)T + (-4.01 - 12.3i)T^{2}
17 1+(−2.99+2.17i)T+(5.25−16.1i)T2 1 + (-2.99 + 2.17i)T + (5.25 - 16.1i)T^{2}
19 1+(2.02+0.659i)T+(15.3+11.1i)T2 1 + (2.02 + 0.659i)T + (15.3 + 11.1i)T^{2}
23 1−6.24iT−23T2 1 - 6.24iT - 23T^{2}
29 1+(−3.09−9.53i)T+(−23.4+17.0i)T2 1 + (-3.09 - 9.53i)T + (-23.4 + 17.0i)T^{2}
31 1+(2.96+2.15i)T+(9.57+29.4i)T2 1 + (2.96 + 2.15i)T + (9.57 + 29.4i)T^{2}
37 1+(2.16+6.66i)T+(−29.9+21.7i)T2 1 + (2.16 + 6.66i)T + (-29.9 + 21.7i)T^{2}
41 1+(−0.0135+0.0416i)T+(−33.1−24.0i)T2 1 + (-0.0135 + 0.0416i)T + (-33.1 - 24.0i)T^{2}
43 1−5.49iT−43T2 1 - 5.49iT - 43T^{2}
47 1+(3.03+0.987i)T+(38.0+27.6i)T2 1 + (3.03 + 0.987i)T + (38.0 + 27.6i)T^{2}
53 1+(−3.00+4.13i)T+(−16.3−50.4i)T2 1 + (-3.00 + 4.13i)T + (-16.3 - 50.4i)T^{2}
59 1+(−11.0+3.59i)T+(47.7−34.6i)T2 1 + (-11.0 + 3.59i)T + (47.7 - 34.6i)T^{2}
61 1+(−2.31−3.19i)T+(−18.8+58.0i)T2 1 + (-2.31 - 3.19i)T + (-18.8 + 58.0i)T^{2}
67 1−6.70T+67T2 1 - 6.70T + 67T^{2}
71 1+(0.527+0.726i)T+(−21.9+67.5i)T2 1 + (0.527 + 0.726i)T + (-21.9 + 67.5i)T^{2}
73 1+(−7.32+2.38i)T+(59.0−42.9i)T2 1 + (-7.32 + 2.38i)T + (59.0 - 42.9i)T^{2}
79 1+(−2.34+3.22i)T+(−24.4−75.1i)T2 1 + (-2.34 + 3.22i)T + (-24.4 - 75.1i)T^{2}
83 1+(−8.76+6.36i)T+(25.6−78.9i)T2 1 + (-8.76 + 6.36i)T + (25.6 - 78.9i)T^{2}
89 1+6.48iT−89T2 1 + 6.48iT - 89T^{2}
97 1+(−2.13−1.55i)T+(29.9+92.2i)T2 1 + (-2.13 - 1.55i)T + (29.9 + 92.2i)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

−11.08925240566920150037804781634, −10.02394984443618977309228538086, −9.499431292760650386554435778928, −8.741246260639576540380398509646, −7.46527326057764489911837943268, −6.61178051614655248456378401230, −5.28194075190092345579561191308, −4.48161629826686457025522139752, −3.42228241832282648531488898417, −2.04738776923316972544956776289, 0.861118362337336697328408735705, 2.34288608605806259539238535444, 3.55668518736839375678354013918, 5.14052084540614745841713641452, 6.05221373707169542028935851208, 6.92136879230498974910031681701, 8.093374609698445159674943225240, 8.471208204089279386756001877136, 9.661094768923171562732979689192, 10.60720578796230748475271333942

Graph of the ZZ-function along the critical line