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Properties

Label 2-3528-504.59-c0-0-7
Degree $2$
Conductor $3528$
Sign $0.00855 + 0.999i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.923 − 0.382i)3-s + (0.499 + 0.866i)4-s + (−0.991 − 0.130i)6-s − 0.999i·8-s + (0.707 − 0.707i)9-s − 0.517i·11-s + (0.793 + 0.608i)12-s + (−0.5 + 0.866i)16-s + (0.991 − 1.71i)17-s + (−0.965 + 0.258i)18-s + (−1.37 + 0.793i)19-s + (−0.258 + 0.448i)22-s + (−0.382 − 0.923i)24-s + 25-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.923 − 0.382i)3-s + (0.499 + 0.866i)4-s + (−0.991 − 0.130i)6-s − 0.999i·8-s + (0.707 − 0.707i)9-s − 0.517i·11-s + (0.793 + 0.608i)12-s + (−0.5 + 0.866i)16-s + (0.991 − 1.71i)17-s + (−0.965 + 0.258i)18-s + (−1.37 + 0.793i)19-s + (−0.258 + 0.448i)22-s + (−0.382 − 0.923i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00855 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00855 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.00855 + 0.999i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.00855 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.183919392\)
\(L(\frac12)\) \(\approx\) \(1.183919392\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 \)
good5 \( 1 - T^{2} \)
11 \( 1 + 0.517iT - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.991 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.37 - 0.793i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.130 - 0.226i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.608 - 1.05i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.05 + 0.608i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.226 - 0.130i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.636549285672250078196473686172, −8.053253398974527155387994325121, −7.21678153484558526971126138660, −6.79689352061657996003463598211, −5.70738494871463650678438761146, −4.44258283539732524067089214356, −3.49203629548336123788179679116, −2.86816960015407869879102883574, −1.98962441070864300376867156640, −0.870031735955594418773138982724, 1.47163751913829746119738372462, 2.30272228364087751792535215107, 3.34998439589904743119744315126, 4.40585108792240170544955354419, 5.12804909888900366899446882387, 6.23658501118992420564189640627, 6.82648429985180779230865008179, 7.71779398495600864158348452201, 8.314933789986285379292945858873, 8.741718104717826425645008229216

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