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Properties

Label 2-528-11.9-c1-0-10
Degree $2$
Conductor $528$
Sign $-0.605 + 0.795i$
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  + (0.809 − 0.587i)3-s + (−0.881 − 2.71i)5-s + (−1.5 − 1.08i)7-s + (0.309 − 0.951i)9-s + (−1.23 + 3.07i)11-s + (1 − 3.07i)13-s + (−2.30 − 1.67i)15-s + (−1.23 − 3.80i)17-s + (−0.618 + 0.449i)19-s − 1.85·21-s − 7.23·23-s + (−2.54 + 1.84i)25-s + (−0.309 − 0.951i)27-s + (−0.309 − 0.224i)29-s + (1.80 − 5.56i)31-s + ⋯
L(s)  = 1  + (0.467 − 0.339i)3-s + (−0.394 − 1.21i)5-s + (−0.566 − 0.411i)7-s + (0.103 − 0.317i)9-s + (−0.372 + 0.927i)11-s + (0.277 − 0.853i)13-s + (−0.596 − 0.433i)15-s + (−0.299 − 0.922i)17-s + (−0.141 + 0.103i)19-s − 0.404·21-s − 1.50·23-s + (−0.509 + 0.369i)25-s + (−0.0594 − 0.183i)27-s + (−0.0573 − 0.0416i)29-s + (0.324 − 0.999i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(4.21610\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.516558 - 1.04196i\)
\(L(\frac12)\) \(\approx\) \(0.516558 - 1.04196i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (1.23 - 3.07i)T \)
good5 \( 1 + (0.881 + 2.71i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (1.5 + 1.08i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1 + 3.07i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.23 + 3.80i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.618 - 0.449i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 7.23T + 23T^{2} \)
29 \( 1 + (0.309 + 0.224i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.80 + 5.56i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.23 - 2.35i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.85 - 4.25i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.70T + 43T^{2} \)
47 \( 1 + (-7.23 + 5.25i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.66 + 11.2i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.35 - 3.16i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.38 - 13.4i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + (3.38 + 10.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-11.6 - 8.45i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.118 + 0.363i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.5 + 1.53i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 6.76T + 89T^{2} \)
97 \( 1 + (-5.02 + 15.4i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.26321980439559474587895760908, −9.662708269411264836242159024620, −8.644781439650956102090081888514, −7.913978401424432070136755498777, −7.13133077569315487768427833327, −5.87122610858418282811492125251, −4.71312349861132037762232836432, −3.80630390177571824941863889406, −2.33348190624159076041923902671, −0.62943214958131817011722912016, 2.32391075580834437084908159024, 3.34381134920930658861509641382, 4.16636837062841361997671357392, 5.85090471039672274165001667871, 6.54334861378031217538741867013, 7.62653761301220814321369654394, 8.542188453796680750810924569139, 9.357423055101189185039952765643, 10.49360943725794649326578052736, 10.88742252357522200530394831499

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