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Properties

Label 2-3528-504.67-c0-0-3
Degree $2$
Conductor $3528$
Sign $0.734 + 0.678i$
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 + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s − i·5-s + 0.999i·6-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s − 11-s + (0.499 − 0.866i)12-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 − 0.499i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s − i·5-s + 0.999i·6-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s − 11-s + (0.499 − 0.866i)12-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 − 0.499i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.734 + 0.678i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.734 + 0.678i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6248114918\)
\(L(\frac12)\) \(\approx\) \(0.6248114918\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.576462577998323874254911353753, −7.972451334442446084198962045925, −7.53827535897046061093112341701, −6.43174177279052988455016045276, −5.89110550955782739909165714035, −4.89163011771293444505764515207, −3.94176903593054202135708550327, −2.76721348129608561514841562916, −1.72611980818439714680252131401, −1.02346558989801407939824780392, 0.65257763591077963300162936042, 2.56192181033021911847629777850, 3.07673739909789942998762493226, 4.45069003149190666810926295700, 5.26829463612457056401339990397, 5.88201720574215753393831276525, 6.83755270224064170392825833244, 7.10228400998945187222309173262, 8.323795427118584936981184762262, 8.741312951073250416682754940909

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