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Properties

Label 2-528-44.43-c1-0-3
Degree $2$
Conductor $528$
Sign $-0.0791 - 0.996i$
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  + i·3-s + 2.73·5-s − 5.13·7-s − 9-s + (1.88 + 2.73i)11-s + 5.13i·13-s + 2.73i·15-s + 3.76i·17-s + 3.76·19-s − 5.13i·21-s + 1.26i·23-s + 2.46·25-s − i·27-s + 2i·31-s + (−2.73 + 1.88i)33-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.22·5-s − 1.94·7-s − 0.333·9-s + (0.566 + 0.823i)11-s + 1.42i·13-s + 0.705i·15-s + 0.912i·17-s + 0.862·19-s − 1.12i·21-s + 0.264i·23-s + 0.492·25-s − 0.192i·27-s + 0.359i·31-s + (−0.475 + 0.327i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $-0.0791 - 0.996i$
Analytic conductor: \(4.21610\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :1/2),\ -0.0791 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.903446 + 0.978003i\)
\(L(\frac12)\) \(\approx\) \(0.903446 + 0.978003i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
11 \( 1 + (-1.88 - 2.73i)T \)
good5 \( 1 - 2.73T + 5T^{2} \)
7 \( 1 + 5.13T + 7T^{2} \)
13 \( 1 - 5.13iT - 13T^{2} \)
17 \( 1 - 3.76iT - 17T^{2} \)
19 \( 1 - 3.76T + 19T^{2} \)
23 \( 1 - 1.26iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 - 0.535T + 37T^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 - 3.76T + 43T^{2} \)
47 \( 1 - 4.19iT - 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 9.46iT - 59T^{2} \)
61 \( 1 + 2.38iT - 61T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 - 10.2iT - 73T^{2} \)
79 \( 1 - 5.13T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 - 5.46T + 97T^{2} \)
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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

−10.80500661928748324966273187158, −9.861901780650944139559865929486, −9.507081394465165023006283540260, −8.973703217072276653309184726314, −7.15569410309150290903651844771, −6.39365909006246876636569323092, −5.72827483457004752827191862035, −4.31501378893847114808764717566, −3.29151701807960160693323846979, −1.92986857107540914392678257284, 0.78097426748412010368823622902, 2.68928646499871485696280740935, 3.36546361051797070805971778504, 5.39255675635810422968123581218, 6.11176204638169074275295459624, 6.69204670312245858379595777662, 7.85385292366142228116700352198, 9.142906844389981402838589704953, 9.657246507533088444846044839782, 10.38143953300174778915351796238

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