↑

Properties

Label 2-3528-504.157-c0-0-2
Degree $2$
Conductor $3528$
Sign $0.592 + 0.805i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + 1.73·5-s + (0.866 − 0.499i)6-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.866 − 1.49i)10-s − 0.999i·12-s + (1.49 + 0.866i)15-s + (−0.5 + 0.866i)16-s + 0.999·18-s + (−0.866 − 1.5i)19-s + (−0.866 − 1.49i)20-s − 23-s + (−0.866 − 0.5i)24-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + 1.73·5-s + (0.866 − 0.499i)6-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.866 − 1.49i)10-s − 0.999i·12-s + (1.49 + 0.866i)15-s + (−0.5 + 0.866i)16-s + 0.999·18-s + (−0.866 − 1.5i)19-s + (−0.866 − 1.49i)20-s − 23-s + (−0.866 − 0.5i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.592 + 0.805i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.592 + 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.718736689\)
\(L(\frac12)\) \(\approx\) \(2.718736689\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 \)
good5 \( 1 - 1.73T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + T + T^{2} \)
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 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.008743372548478316186140320525, −8.284434855770687540672568023377, −6.97025898866061425636408602262, −6.19523619932347851100429118368, −5.41886377357936195570998954984, −4.72533628252201551206491413162, −3.94979833021120617646471566025, −2.77317902782277587216845953945, −2.36060904184755265515560302030, −1.49284217269098366485974919895, 1.66370152428346974851814850829, 2.38682055650684543456572838803, 3.39510869976281655838227019726, 4.27100168184256196346688372650, 5.31918430281186626348800863591, 6.16337252331402888312046417073, 6.35410251340992537824381783918, 7.32595216136246632854264020959, 8.148751356093869309253502945305, 8.629463124952444168815901111540

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