Properties

Degree $2$
Conductor $3528$
Sign $0.827 - 0.561i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 0.999·8-s + (−0.5 − 0.866i)16-s + (−0.707 + 1.22i)17-s + (0.707 + 1.22i)19-s + (−0.5 + 0.866i)25-s + (−0.499 + 0.866i)32-s + 1.41·34-s + (0.707 − 1.22i)38-s − 1.41·41-s + 0.999·50-s + (−0.707 + 1.22i)59-s + 0.999·64-s + (1 − 1.73i)67-s + (−0.707 − 1.22i)68-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 0.999·8-s + (−0.5 − 0.866i)16-s + (−0.707 + 1.22i)17-s + (0.707 + 1.22i)19-s + (−0.5 + 0.866i)25-s + (−0.499 + 0.866i)32-s + 1.41·34-s + (0.707 − 1.22i)38-s − 1.41·41-s + 0.999·50-s + (−0.707 + 1.22i)59-s + 0.999·64-s + (1 − 1.73i)67-s + (−0.707 − 1.22i)68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.827 - 0.561i$
Motivic weight: \(0\)
Character: $\chi_{3528} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.827 - 0.561i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7396200125\)
\(L(\frac12)\) \(\approx\) \(0.7396200125\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
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 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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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

−8.896443951494270257944402578214, −8.176245117371032844428910751283, −7.63831594265698707798775227941, −6.71157398513552065834838634133, −5.77763032657000889302325771798, −4.86681238347315266997285581112, −3.87247669782138582780345235465, −3.37248104038385488029820420339, −2.14987089495500068427035132349, −1.37156773131665540528566931529, 0.53661693225407695761098432685, 1.98523321486003270273318274431, 3.10589249017634638689089847983, 4.41086108454520350672420838128, 4.96070999035228669709231458711, 5.75633845321950685573796082268, 6.72462101648917847911808398627, 7.07479430306699047316323794013, 7.938948155557544694554147825907, 8.635271897428709529382489497989

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