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Properties

Label 2-528-44.43-c5-0-34
Degree $2$
Conductor $528$
Sign $0.170 + 0.985i$
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  + 9i·3-s − 34.2·5-s − 120.·7-s − 81·9-s + (−257. − 308. i)11-s + 1.15e3i·13-s − 308. i·15-s + 1.40e3i·17-s + 903.·19-s − 1.08e3i·21-s + 5.01e3i·23-s − 1.94e3·25-s − 729i·27-s − 2.34e3i·29-s + 6.18e3i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.613·5-s − 0.930·7-s − 0.333·9-s + (−0.640 − 0.767i)11-s + 1.90i·13-s − 0.354i·15-s + 1.17i·17-s + 0.574·19-s − 0.537i·21-s + 1.97i·23-s − 0.623·25-s − 0.192i·27-s − 0.516i·29-s + 1.15i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $0.170 + 0.985i$
Analytic conductor: \(84.6826\)
Root analytic conductor: \(9.20231\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :5/2),\ 0.170 + 0.985i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.05405473573\)
\(L(\frac12)\) \(\approx\) \(0.05405473573\)
\(L(\frac{7}{2})\) 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 \( 1 - 9iT \)
11 \( 1 + (257. + 308. i)T \)
good5 \( 1 + 34.2T + 3.12e3T^{2} \)
7 \( 1 + 120.T + 1.68e4T^{2} \)
13 \( 1 - 1.15e3iT - 3.71e5T^{2} \)
17 \( 1 - 1.40e3iT - 1.41e6T^{2} \)
19 \( 1 - 903.T + 2.47e6T^{2} \)
23 \( 1 - 5.01e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.34e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.18e3iT - 2.86e7T^{2} \)
37 \( 1 + 6.25e3T + 6.93e7T^{2} \)
41 \( 1 + 6.01e3iT - 1.15e8T^{2} \)
43 \( 1 + 3.33e3T + 1.47e8T^{2} \)
47 \( 1 + 2.57e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.02e4T + 4.18e8T^{2} \)
59 \( 1 + 9.81e3iT - 7.14e8T^{2} \)
61 \( 1 - 1.72e3iT - 8.44e8T^{2} \)
67 \( 1 - 2.56e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.92e4iT - 1.80e9T^{2} \)
73 \( 1 + 6.13e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.21e4T + 3.07e9T^{2} \)
83 \( 1 - 3.65e4T + 3.93e9T^{2} \)
89 \( 1 + 3.22e4T + 5.58e9T^{2} \)
97 \( 1 + 4.75e4T + 8.58e9T^{2} \)
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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

−9.789809522621827271229645100633, −9.096493193216136573985400000071, −8.183121195475266723447538891042, −7.11978990260076558569372618319, −6.16441696184170184447130498383, −5.15879505937949095036282301221, −3.83135696876018805047553285904, −3.40706767352663712223828925365, −1.78279063621695410960308079394, −0.01782053496486479184581014850, 0.70911048082060631190539906186, 2.54512005327009747865231340669, 3.22552106827312005019548136000, 4.67692804006757641797060401701, 5.69711455209011946993461857574, 6.75292316180219556991685692906, 7.63514680958501531286067676705, 8.179976440207659406623456611531, 9.481441200548712454687170458819, 10.22078918772773275893844060202

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