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Properties

Label 2-528-44.43-c5-0-0
Degree $2$
Conductor $528$
Sign $-0.906 - 0.421i$
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 + 57.6·5-s − 131.·7-s − 81·9-s + (230. + 328. i)11-s + 558. i·13-s − 519. i·15-s − 2.07e3i·17-s + 347.·19-s + 1.18e3i·21-s + 3.83e3i·23-s + 203.·25-s + 729i·27-s − 4.84e3i·29-s + 852. i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.03·5-s − 1.01·7-s − 0.333·9-s + (0.574 + 0.818i)11-s + 0.916i·13-s − 0.595i·15-s − 1.74i·17-s + 0.220·19-s + 0.584i·21-s + 1.51i·23-s + 0.0649·25-s + 0.192i·27-s − 1.06i·29-s + 0.159i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $-0.906 - 0.421i$
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: \((2,\ 528,\ (\ :5/2),\ -0.906 - 0.421i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.07649700594\)
\(L(\frac12)\) \(\approx\) \(0.07649700594\)
\(L(\frac{7}{2})\) 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 \( 1 + (-230. - 328. i)T \)
good5 \( 1 - 57.6T + 3.12e3T^{2} \)
7 \( 1 + 131.T + 1.68e4T^{2} \)
13 \( 1 - 558. iT - 3.71e5T^{2} \)
17 \( 1 + 2.07e3iT - 1.41e6T^{2} \)
19 \( 1 - 347.T + 2.47e6T^{2} \)
23 \( 1 - 3.83e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.84e3iT - 2.05e7T^{2} \)
31 \( 1 - 852. iT - 2.86e7T^{2} \)
37 \( 1 + 1.46e4T + 6.93e7T^{2} \)
41 \( 1 + 1.49e4iT - 1.15e8T^{2} \)
43 \( 1 + 2.14e3T + 1.47e8T^{2} \)
47 \( 1 + 8.29e3iT - 2.29e8T^{2} \)
53 \( 1 + 3.37e4T + 4.18e8T^{2} \)
59 \( 1 - 741. iT - 7.14e8T^{2} \)
61 \( 1 - 4.81e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.81e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.40e4iT - 1.80e9T^{2} \)
73 \( 1 + 4.53e3iT - 2.07e9T^{2} \)
79 \( 1 - 4.47e4T + 3.07e9T^{2} \)
83 \( 1 + 8.16e4T + 3.93e9T^{2} \)
89 \( 1 + 2.82e4T + 5.58e9T^{2} \)
97 \( 1 - 4.34e4T + 8.58e9T^{2} \)
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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

−10.13192962223030187475348534795, −9.485016279945362727860167821839, −9.041316639185784558441190133952, −7.39476446524206323776340796946, −6.87091228641885631922465153461, −5.99318500769244814186573482195, −5.01817858225657635719987063317, −3.58505842186329635060298992947, −2.36620540953107047100179217676, −1.47321355294492199578254136864, 0.01596570787332101493530748129, 1.49516358788686771399691623777, 2.94379614358230268610928054100, 3.70742218883218131164327937399, 5.09958901069394789971273285338, 6.14297746044136077122127900787, 6.46902022250008059936876962986, 8.168633894565527858691064466225, 8.903689319149686994598192248921, 9.819086540522042501353524210868

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