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Properties

Label 2-3528-72.43-c0-0-5
Degree $2$
Conductor $3528$
Sign $-0.984 - 0.173i$
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 + (−0.866 + 0.5i)5-s + 0.999i·6-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s + (0.5 − 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s + 17-s + (0.866 − 0.499i)18-s − 19-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + 0.999i·6-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s + (0.5 − 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s + 17-s + (0.866 − 0.499i)18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.984 - 0.173i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2022188474\)
\(L(\frac12)\) \(\approx\) \(0.2022188474\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - iT - T^{2} \)
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 \( 1 + iT - T^{2} \)
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 \( 1 + T + T^{2} \)
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 \( 1 + T + T^{2} \)
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

−8.369692084829507249966076536939, −7.56623737658236347202436486844, −7.09624325762129458868040765793, −6.47560255494082344491196512664, −5.53180956507558771160557584707, −4.29931682151293741919426009340, −3.39864557703299118438646358323, −2.55651692164476565764715583662, −1.49614180284212668885195459786, −0.18575091203530133148403206079, 1.27297274609380430027497668110, 2.77811340387787122638099571724, 3.96514661201432798844170104391, 4.70525641638961301599220273962, 5.37087692657741975151146833972, 6.15801425521371923824834134501, 7.15265739055284454859344791009, 7.63360244313688579326072212944, 8.490621689289729590995029150776, 9.174356066945332835880687283474

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