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Properties

Label 2-3528-56.13-c0-0-3
Degree $2$
Conductor $3528$
Sign $0.654 - 0.755i$
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  + i·2-s − 4-s + 1.73·5-s − i·8-s + 1.73i·10-s − i·11-s + 16-s − 1.73·20-s + 22-s + 1.99·25-s + i·29-s − 1.73i·31-s + i·32-s − 1.73i·40-s + i·44-s + ⋯
L(s)  = 1  + i·2-s − 4-s + 1.73·5-s − i·8-s + 1.73i·10-s − i·11-s + 16-s − 1.73·20-s + 22-s + 1.99·25-s + i·29-s − 1.73i·31-s + i·32-s − 1.73i·40-s + i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.654 - 0.755i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.654 - 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.587741419\)
\(L(\frac12)\) \(\approx\) \(1.587741419\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 1.73T + T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 - 1.73T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + 1.73T + T^{2} \)
89 \( 1 - T^{2} \)
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

−8.855367932406073764821197033003, −8.176826574185748723248119962229, −7.21201317792271912714305854405, −6.48069265154468218136253417991, −5.80513005592727699718192273727, −5.48979427799134967975763327924, −4.52528228267829412966910626289, −3.43243593251433773561261784100, −2.36811570449277919607665540668, −1.11047489824872060912961662568, 1.33338235996394542513060076790, 2.10249549315054070032698643319, 2.77191738708926786950871093926, 3.91707084961185202064818833665, 4.93600730079001543264896680910, 5.38057765933395831952489489160, 6.30652242895928354825872792437, 7.06948644922288458426827026949, 8.199718815188268912435577511211, 8.951939027419018178763938646644

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