Properties

Degree $2$
Conductor $3528$
Sign $0.997 + 0.0633i$
Motivic weight $0$
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 + (0.499 − 0.866i)4-s + (0.866 + 1.5i)5-s − 0.999i·8-s + (1.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)16-s + 1.73·20-s + 0.999·22-s + (−1 + 1.73i)25-s + i·29-s + (−1.5 − 0.866i)31-s + (−0.866 − 0.499i)32-s + (1.49 − 0.866i)40-s + (0.866 − 0.499i)44-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.866 + 1.5i)5-s − 0.999i·8-s + (1.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)16-s + 1.73·20-s + 0.999·22-s + (−1 + 1.73i)25-s + i·29-s + (−1.5 − 0.866i)31-s + (−0.866 − 0.499i)32-s + (1.49 − 0.866i)40-s + (0.866 − 0.499i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.997 + 0.0633i$
Motivic weight: \(0\)
Character: $\chi_{3528} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.997 + 0.0633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.561816920\)
\(L(\frac12)\) \(\approx\) \(2.561816920\)
\(L(1)\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - 1.73T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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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

−9.152677016315313831018395252183, −7.67906435256990149148093352557, −6.89337996629614449355205227455, −6.51115755581602765917300600623, −5.75898325855555291322067703194, −4.98631094857944052906232788493, −3.84982242891029496866036603075, −3.28503536118662693449654919080, −2.29776339363999742878148785697, −1.65469086099711228253362739435, 1.30068320453542783647605866453, 2.27172396268634410937145333381, 3.52796638331368601987844280031, 4.31656233748741455226282786416, 5.03751937344618109071226814587, 5.73354281641219609440268816672, 6.22331809369899560425638436067, 7.15014460247115465133257065028, 8.043639191691537781513764962611, 8.804970505731845945680269475825

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