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Properties

Label 2-3528-392.125-c0-0-0
Degree $2$
Conductor $3528$
Sign $0.926 - 0.375i$
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  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.193 − 0.846i)5-s + (−0.222 − 0.974i)7-s + (−0.974 − 0.222i)8-s + (0.846 − 0.193i)10-s + (0.541 + 1.12i)11-s + (0.781 − 0.623i)14-s + (−0.222 − 0.974i)16-s + (0.541 + 0.678i)20-s + (−0.777 + 0.974i)22-s + (0.222 + 0.107i)25-s + (0.900 + 0.433i)28-s + (1.40 − 1.12i)29-s − 1.94i·31-s + (0.781 − 0.623i)32-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.193 − 0.846i)5-s + (−0.222 − 0.974i)7-s + (−0.974 − 0.222i)8-s + (0.846 − 0.193i)10-s + (0.541 + 1.12i)11-s + (0.781 − 0.623i)14-s + (−0.222 − 0.974i)16-s + (0.541 + 0.678i)20-s + (−0.777 + 0.974i)22-s + (0.222 + 0.107i)25-s + (0.900 + 0.433i)28-s + (1.40 − 1.12i)29-s − 1.94i·31-s + (0.781 − 0.623i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.926 - 0.375i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.926 - 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.502729332\)
\(L(\frac12)\) \(\approx\) \(1.502729332\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.433 - 0.900i)T \)
3 \( 1 \)
7 \( 1 + (0.222 + 0.974i)T \)
good5 \( 1 + (-0.193 + 0.846i)T + (-0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.541 - 1.12i)T + (-0.623 + 0.781i)T^{2} \)
13 \( 1 + (0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.222 - 0.974i)T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.222 - 0.974i)T^{2} \)
29 \( 1 + (-1.40 + 1.12i)T + (0.222 - 0.974i)T^{2} \)
31 \( 1 + 1.94iT - T^{2} \)
37 \( 1 + (0.222 - 0.974i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (-1.56 - 1.24i)T + (0.222 + 0.974i)T^{2} \)
59 \( 1 + (-0.900 + 0.433i)T^{2} \)
61 \( 1 + (-0.222 + 0.974i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.846 + 1.75i)T + (-0.623 - 0.781i)T^{2} \)
79 \( 1 - 0.445T + T^{2} \)
83 \( 1 + (-0.781 - 0.376i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.623 - 0.781i)T^{2} \)
97 \( 1 + 1.56iT - T^{2} \)
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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

−8.665299418752631944434864300281, −7.86854431581737041682165029058, −7.31307631696430860441137963471, −6.55032241723677504307022348627, −5.89852667128194701797790139800, −4.84085136285845607421408380143, −4.39250357269709809794190844371, −3.73284691530471343820472450027, −2.40622912982353829128922000849, −0.905173479951639148071386051799, 1.23990138328281876009946041289, 2.48287025638223140086078280845, 3.07888894101403832649915186550, 3.72709184483644822331654919368, 4.98076867672780404985658289451, 5.54499249989146292641485466570, 6.48604826945366230895502316808, 6.79529359804729337948407726883, 8.456463592717712572055569693144, 8.668748480650094934772638393798

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