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Properties

Label 2-528-11.3-c1-0-7
Degree $2$
Conductor $528$
Sign $0.834 + 0.551i$
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  + (−0.309 − 0.951i)3-s + (1.30 + 0.951i)5-s + (−0.0729 + 0.224i)7-s + (−0.809 + 0.587i)9-s + (0.809 − 3.21i)11-s + (3.42 − 2.48i)13-s + (0.499 − 1.53i)15-s + (0.118 + 0.0857i)17-s + (2.11 + 6.51i)19-s + 0.236·21-s + 5·23-s + (−0.736 − 2.26i)25-s + (0.809 + 0.587i)27-s + (0.618 − 1.90i)29-s + (2.73 − 1.98i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (0.585 + 0.425i)5-s + (−0.0275 + 0.0848i)7-s + (−0.269 + 0.195i)9-s + (0.243 − 0.969i)11-s + (0.950 − 0.690i)13-s + (0.129 − 0.397i)15-s + (0.0286 + 0.0207i)17-s + (0.485 + 1.49i)19-s + 0.0515·21-s + 1.04·23-s + (−0.147 − 0.453i)25-s + (0.155 + 0.113i)27-s + (0.114 − 0.353i)29-s + (0.491 − 0.357i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $0.834 + 0.551i$
Analytic conductor: \(4.21610\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :1/2),\ 0.834 + 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49662 - 0.450156i\)
\(L(\frac12)\) \(\approx\) \(1.49662 - 0.450156i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.809 + 3.21i)T \)
good5 \( 1 + (-1.30 - 0.951i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.0729 - 0.224i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.42 + 2.48i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.118 - 0.0857i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.11 - 6.51i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5T + 23T^{2} \)
29 \( 1 + (-0.618 + 1.90i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.73 + 1.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.690 + 2.12i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.69 + 8.28i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.52T + 43T^{2} \)
47 \( 1 + (2.95 + 9.09i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.35 - 3.88i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.64 - 11.2i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-8.78 - 6.37i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.38T + 67T^{2} \)
71 \( 1 + (11.7 + 8.55i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.85 - 14.9i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.04 - 5.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.42 + 1.03i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 5.18T + 89T^{2} \)
97 \( 1 + (12.1 - 8.83i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.73547275145947364181234771049, −10.08124489331243158531221118454, −8.847206689438132349695702323911, −8.131299794303231184660347493953, −7.06121320135829142292749847952, −5.96934643472988870647415420071, −5.64098103540692299657699111564, −3.82820204515274875978879110040, −2.70706353928177007759572660854, −1.16481504670238952861393537666, 1.44583843192524695666275196921, 3.07802707238408678609122230808, 4.47235306557935738485096643253, 5.11680339831629015324186286983, 6.35253735304241426841730641149, 7.14671588223886397448961342554, 8.548634553396841733906651259541, 9.327461875385622003763270710195, 9.834709219316129492509996481986, 11.03680299065074745378352963401

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