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Properties

Label 2-3528-72.67-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: $\chi_{3528} (3235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.206655742\)
\(L(\frac12)\) \(\approx\) \(2.206655742\)
\(L(1)\) 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

−9.125021917279018341254423490705, −8.097802057351498999377959655898, −6.67351770345566258093606844018, −6.36308217805734846876228010657, −5.79921657306849615897391611616, −4.76676792954973029526499125662, −4.25140011418134892422980319928, −3.40985955572413424981345275680, −2.43861747304447788542711629938, −1.42576062837841225908658133342, 1.23217309827898492178071716728, 2.19186759521768043955747887430, 3.30277537054758196906221565016, 4.19112552664528034720605501589, 5.50809492341416810523843782690, 5.59492169770929319356041996713, 6.28699124025639060076281459193, 7.00670554992198859342593217097, 7.931718409429084214182652133394, 8.464924875520780852469959319345

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