Properties

Label 2-3528-8.3-c0-0-1
Degree $2$
Conductor $3528$
Sign $1$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 2·11-s + 16-s − 2·22-s + 25-s − 32-s − 2·43-s + 2·44-s − 50-s + 64-s + 2·67-s + 2·86-s − 2·88-s + 100-s − 2·107-s + 2·113-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s + 2·11-s + 16-s − 2·22-s + 25-s − 32-s − 2·43-s + 2·44-s − 50-s + 64-s + 2·67-s + 2·86-s − 2·88-s + 100-s − 2·107-s + 2·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3528} (883, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9479227530\)
\(L(\frac12)\) \(\approx\) \(0.9479227530\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.706261240350169393988595552389, −8.285981089508887176764867615008, −7.17570084857949925488445180970, −6.70778009095661027703257213759, −6.11102713147333405931398758047, −5.03051945112957078558261094584, −3.91623062565850856913642172481, −3.15156071375818914698984792631, −1.93212856658993978577026009985, −1.05447110805962188879110148449, 1.05447110805962188879110148449, 1.93212856658993978577026009985, 3.15156071375818914698984792631, 3.91623062565850856913642172481, 5.03051945112957078558261094584, 6.11102713147333405931398758047, 6.70778009095661027703257213759, 7.17570084857949925488445180970, 8.285981089508887176764867615008, 8.706261240350169393988595552389

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