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Properties

Label 2-528-33.2-c1-0-5
Degree $2$
Conductor $528$
Sign $0.743 - 0.668i$
Analytic cond. $4.21610$
Root an. cond. $2.05331$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $0.743 - 0.668i$
Analytic conductor: \(4.21610\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :1/2),\ 0.743 - 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.902905 + 0.346245i\)
\(L(\frac12)\) \(\approx\) \(0.902905 + 0.346245i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.809 + 1.53i)T \)
11 \( 1 + (0.151 - 3.31i)T \)
good5 \( 1 + (0.897 - 0.291i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (0.897 - 1.23i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-4.03 - 1.30i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.906 + 2.78i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.16 - 5.73i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.18iT - 23T^{2} \)
29 \( 1 + (-5.17 - 3.75i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.68 + 8.27i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.28 + 1.66i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.92 - 4.30i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.34iT - 43T^{2} \)
47 \( 1 + (-5.58 - 7.68i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-4.62 - 1.50i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.74 - 3.77i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.685 - 0.222i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 3.89T + 67T^{2} \)
71 \( 1 + (9.47 - 3.07i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.03 + 1.41i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.56 - 0.508i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.90 + 5.85i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 17.3iT - 89T^{2} \)
97 \( 1 + (-3.54 + 10.8i)T + (-78.4 - 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line