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Properties

Label 2-3528-504.221-c0-0-0
Degree $2$
Conductor $3528$
Sign $0.220 - 0.975i$
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 + (−0.258 − 0.965i)3-s + (0.499 + 0.866i)4-s + 0.517·5-s + (−0.258 + 0.965i)6-s − 0.999i·8-s + (−0.866 + 0.499i)9-s + (−0.448 − 0.258i)10-s + (0.707 − 0.707i)12-s + (−1.22 − 0.707i)13-s + (−0.133 − 0.499i)15-s + (−0.5 + 0.866i)16-s + 18-s + (−1.67 + 0.965i)19-s + (0.258 + 0.448i)20-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.258 − 0.965i)3-s + (0.499 + 0.866i)4-s + 0.517·5-s + (−0.258 + 0.965i)6-s − 0.999i·8-s + (−0.866 + 0.499i)9-s + (−0.448 − 0.258i)10-s + (0.707 − 0.707i)12-s + (−1.22 − 0.707i)13-s + (−0.133 − 0.499i)15-s + (−0.5 + 0.866i)16-s + 18-s + (−1.67 + 0.965i)19-s + (0.258 + 0.448i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.220 - 0.975i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.220 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2010822384\)
\(L(\frac12)\) \(\approx\) \(0.2010822384\)
\(L(1)\) 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 \( 1 + (0.258 + 0.965i)T \)
7 \( 1 \)
good5 \( 1 - 0.517T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 - 1.73iT - T^{2} \)
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 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−8.858972160100531165013693842262, −8.067486781223945441070610686630, −7.60281509736708256695996040158, −6.89742185298045557329324974636, −6.04091417209974944777494880863, −5.42268355991125822858157855632, −4.12819538770072394355918270175, −2.99192377272129090647285200751, −2.16273542623887163829141729809, −1.46600201501726075882813658807, 0.14931249418387724665983649235, 2.05186686508014824496742667289, 2.72889447356185529456384418447, 4.35306993054698309078055104597, 4.75253037185458579306638168809, 5.71946341564386473881583932622, 6.45159645130228892616749715268, 6.94933358082891600370776689212, 8.067097857629716400618964344335, 8.772614765154264366484456550201

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