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Properties

Label 2-528-33.2-c1-0-5
Degree 22
Conductor 528528
Sign 0.743−0.668i0.743 - 0.668i
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  + (−0.809 − 1.53i)3-s + (−0.897 + 0.291i)5-s + (−0.897 + 1.23i)7-s + (−1.69 + 2.47i)9-s + (−0.151 + 3.31i)11-s + (4.03 + 1.30i)13-s + (1.17 + 1.13i)15-s + (−0.906 − 2.78i)17-s + (4.16 + 5.73i)19-s + (2.61 + 0.375i)21-s − 1.18i·23-s + (−3.32 + 2.41i)25-s + (5.16 + 0.584i)27-s + (5.17 + 3.75i)29-s + (2.68 − 8.27i)31-s + ⋯
L(s)  = 1  + (−0.467 − 0.884i)3-s + (−0.401 + 0.130i)5-s + (−0.339 + 0.466i)7-s + (−0.563 + 0.826i)9-s + (−0.0457 + 0.998i)11-s + (1.11 + 0.363i)13-s + (0.302 + 0.293i)15-s + (−0.219 − 0.676i)17-s + (0.956 + 1.31i)19-s + (0.571 + 0.0818i)21-s − 0.247i·23-s + (−0.664 + 0.483i)25-s + (0.993 + 0.112i)27-s + (0.960 + 0.697i)29-s + (0.482 − 1.48i)31-s + ⋯

Functional equation

Λ(s)=(528s/2 ΓC(s) L(s)=((0.743−0.668i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(528s/2 ΓC(s+1/2) L(s)=((0.743−0.668i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 528528    =    24⋅3⋅112^{4} \cdot 3 \cdot 11
Sign: 0.743−0.668i0.743 - 0.668i
Analytic conductor: 4.216104.21610
Root analytic conductor: 2.053312.05331
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ528(497,⋅)\chi_{528} (497, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 528, ( :1/2), 0.743−0.668i)(2,\ 528,\ (\ :1/2),\ 0.743 - 0.668i)

Particular Values

L(1)L(1) ≈\approx 0.902905+0.346245i0.902905 + 0.346245i
L(12)L(\frac12) ≈\approx 0.902905+0.346245i0.902905 + 0.346245i
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+(0.809+1.53i)T 1 + (0.809 + 1.53i)T
11 1+(0.151−3.31i)T 1 + (0.151 - 3.31i)T
good5 1+(0.897−0.291i)T+(4.04−2.93i)T2 1 + (0.897 - 0.291i)T + (4.04 - 2.93i)T^{2}
7 1+(0.897−1.23i)T+(−2.16−6.65i)T2 1 + (0.897 - 1.23i)T + (-2.16 - 6.65i)T^{2}
13 1+(−4.03−1.30i)T+(10.5+7.64i)T2 1 + (-4.03 - 1.30i)T + (10.5 + 7.64i)T^{2}
17 1+(0.906+2.78i)T+(−13.7+9.99i)T2 1 + (0.906 + 2.78i)T + (-13.7 + 9.99i)T^{2}
19 1+(−4.16−5.73i)T+(−5.87+18.0i)T2 1 + (-4.16 - 5.73i)T + (-5.87 + 18.0i)T^{2}
23 1+1.18iT−23T2 1 + 1.18iT - 23T^{2}
29 1+(−5.17−3.75i)T+(8.96+27.5i)T2 1 + (-5.17 - 3.75i)T + (8.96 + 27.5i)T^{2}
31 1+(−2.68+8.27i)T+(−25.0−18.2i)T2 1 + (-2.68 + 8.27i)T + (-25.0 - 18.2i)T^{2}
37 1+(2.28+1.66i)T+(11.4+35.1i)T2 1 + (2.28 + 1.66i)T + (11.4 + 35.1i)T^{2}
41 1+(5.92−4.30i)T+(12.6−38.9i)T2 1 + (5.92 - 4.30i)T + (12.6 - 38.9i)T^{2}
43 1−5.34iT−43T2 1 - 5.34iT - 43T^{2}
47 1+(−5.58−7.68i)T+(−14.5+44.6i)T2 1 + (-5.58 - 7.68i)T + (-14.5 + 44.6i)T^{2}
53 1+(−4.62−1.50i)T+(42.8+31.1i)T2 1 + (-4.62 - 1.50i)T + (42.8 + 31.1i)T^{2}
59 1+(2.74−3.77i)T+(−18.2−56.1i)T2 1 + (2.74 - 3.77i)T + (-18.2 - 56.1i)T^{2}
61 1+(0.685−0.222i)T+(49.3−35.8i)T2 1 + (0.685 - 0.222i)T + (49.3 - 35.8i)T^{2}
67 1+3.89T+67T2 1 + 3.89T + 67T^{2}
71 1+(9.47−3.07i)T+(57.4−41.7i)T2 1 + (9.47 - 3.07i)T + (57.4 - 41.7i)T^{2}
73 1+(−1.03+1.41i)T+(−22.5−69.4i)T2 1 + (-1.03 + 1.41i)T + (-22.5 - 69.4i)T^{2}
79 1+(−1.56−0.508i)T+(63.9+46.4i)T2 1 + (-1.56 - 0.508i)T + (63.9 + 46.4i)T^{2}
83 1+(1.90+5.85i)T+(−67.1+48.7i)T2 1 + (1.90 + 5.85i)T + (-67.1 + 48.7i)T^{2}
89 1−17.3iT−89T2 1 - 17.3iT - 89T^{2}
97 1+(−3.54+10.8i)T+(−78.4−57.0i)T2 1 + (-3.54 + 10.8i)T + (-78.4 - 57.0i)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

−11.19925146335010074443416222526, −10.12862636627393807342535832703, −9.149813418860122259142072563437, −8.047164046683119181993288893224, −7.35315563707262449283318018887, −6.38032785459541258910770861232, −5.60252802995582398500129309689, −4.31592464711625650931881090512, −2.87714364244171374790092873417, −1.45672439881641181230426113871, 0.65938177027641450999057555364, 3.19853305447220501358837363897, 3.90596460773039279353205444796, 5.08457779162059283325102500776, 6.03498971480284268892011209257, 6.94827943242707990444058812026, 8.403152525472750631106740881143, 8.855469022834922351116212144989, 10.15066948493059845081024955306, 10.63759488461104650097739637079

Graph of the ZZ-function along the critical line