↑

Properties

Label 2-3528-504.157-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.6249821604\)
\(L(\frac12)\) \(\approx\) \(0.6249821604\)
\(L(1)\) 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

−8.410745173582519533786106672604, −7.925299726716795285063323844343, −7.16662839425658632759642954649, −6.26769287835911802236153452383, −5.48816836574634368656535341348, −4.69738356936849020945953159931, −3.93420411185966584091009625722, −3.34264099669309637269865158455, −1.99725985665076023628412789238, −0.828639350533441648242716140987, 0.52295272085902699791821479950, 2.96386239084847630046122890208, 3.76637709384752569439902019718, 4.37909799265765769939258035681, 4.97814303889992368844988227261, 5.78199537281687662420233266465, 6.75412352287600856274613734225, 7.23084059806910876805250247301, 7.907702927992983456234008045845, 8.696135051412915919202602508278

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