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Properties

Label 2-528-12.11-c5-0-31
Degree $2$
Conductor $528$
Sign $-0.918 + 0.394i$
Analytic cond. $84.6826$
Root an. cond. $9.20231$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (12.4 + 9.33i)3-s + 75.8i·5-s + 181. i·7-s + (68.8 + 233. i)9-s − 121·11-s − 176.·13-s + (−707. + 946. i)15-s + 1.03e3i·17-s − 373. i·19-s + (−1.69e3 + 2.26e3i)21-s − 416.·23-s − 2.62e3·25-s + (−1.31e3 + 3.55e3i)27-s − 1.87e3i·29-s + 9.31e3i·31-s + ⋯
L(s)  = 1  + (0.801 + 0.598i)3-s + 1.35i·5-s + 1.40i·7-s + (0.283 + 0.958i)9-s − 0.301·11-s − 0.289·13-s + (−0.811 + 1.08i)15-s + 0.869i·17-s − 0.237i·19-s + (−0.839 + 1.12i)21-s − 0.163·23-s − 0.839·25-s + (−0.346 + 0.937i)27-s − 0.413i·29-s + 1.74i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $-0.918 + 0.394i$
Analytic conductor: \(84.6826\)
Root analytic conductor: \(9.20231\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :5/2),\ -0.918 + 0.394i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.393716131\)
\(L(\frac12)\) \(\approx\) \(2.393716131\)
\(L(\frac{7}{2})\) 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 \( 1 + (-12.4 - 9.33i)T \)
11 \( 1 + 121T \)
good5 \( 1 - 75.8iT - 3.12e3T^{2} \)
7 \( 1 - 181. iT - 1.68e4T^{2} \)
13 \( 1 + 176.T + 3.71e5T^{2} \)
17 \( 1 - 1.03e3iT - 1.41e6T^{2} \)
19 \( 1 + 373. iT - 2.47e6T^{2} \)
23 \( 1 + 416.T + 6.43e6T^{2} \)
29 \( 1 + 1.87e3iT - 2.05e7T^{2} \)
31 \( 1 - 9.31e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.02e3T + 6.93e7T^{2} \)
41 \( 1 - 8.65e3iT - 1.15e8T^{2} \)
43 \( 1 + 4.73e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.25e4T + 2.29e8T^{2} \)
53 \( 1 + 3.29e4iT - 4.18e8T^{2} \)
59 \( 1 - 6.61e3T + 7.14e8T^{2} \)
61 \( 1 - 1.14e4T + 8.44e8T^{2} \)
67 \( 1 + 6.54e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.45e4T + 1.80e9T^{2} \)
73 \( 1 + 1.21e4T + 2.07e9T^{2} \)
79 \( 1 - 4.34e4iT - 3.07e9T^{2} \)
83 \( 1 + 2.10e4T + 3.93e9T^{2} \)
89 \( 1 + 2.02e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.61e4T + 8.58e9T^{2} \)
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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

−10.47061737395212832903573977317, −9.720644203273061747834231880823, −8.783330364211705974733739836222, −8.053791654660540585566418368053, −6.99358787111914306608735759022, −5.97842491726052905649856108623, −4.93780770527380643181674546862, −3.58193543125113949105504607356, −2.74597078697909828000175140020, −2.04945962108599871274579543863, 0.50904485441991907690586118564, 1.12066249363328095727312158252, 2.45357147163587200599740441047, 3.87553772528008874267789799680, 4.58892145703995782740463210959, 5.86169435421085846099906232685, 7.24513782392631490916406560765, 7.66163830333989909293557601315, 8.647510290895839213368890646677, 9.423954749098286039965507862248

Graph of the $Z$-function along the critical line