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Properties

Label 2-3528-3.2-c0-0-3
Degree $2$
Conductor $3528$
Sign $-0.816 + 0.577i$
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  − 1.41i·5-s − 1.41i·11-s − 13-s − 19-s − 1.00·25-s − 31-s + 37-s − 1.41i·41-s − 43-s + 1.41i·47-s − 2.00·55-s + 1.41i·65-s − 67-s + 1.41i·71-s + 73-s + ⋯
L(s)  = 1  − 1.41i·5-s − 1.41i·11-s − 13-s − 19-s − 1.00·25-s − 31-s + 37-s − 1.41i·41-s − 43-s + 1.41i·47-s − 2.00·55-s + 1.41i·65-s − 67-s + 1.41i·71-s + 73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.816 + 0.577i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.816 + 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8349895706\)
\(L(\frac12)\) \(\approx\) \(0.8349895706\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
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

−8.484203178123183192353498170526, −7.946776818528765003728186015330, −7.01577948241989213884263781450, −6.00731598375258856756709413835, −5.43540895051689889280931443875, −4.65674984675802056763699125099, −3.93163565521617200024834074640, −2.83693049799606661499954259705, −1.69635927657562640312497724338, −0.45868234328069865893421138394, 1.95146520649301826951840366306, 2.54109836783216901845155411096, 3.52989060463489979234613393059, 4.45604792239740404381996476053, 5.20220034525360588401754815157, 6.37456668973445692327809245241, 6.80349605717747208566743382481, 7.47276049556730283720575287328, 8.071865626468637297677640735748, 9.249313022419241038025943340773

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