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Properties

Label 2-3528-504.229-c0-0-0
Degree $2$
Conductor $3528$
Sign $0.971 - 0.235i$
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  − 2-s + (−0.866 + 0.5i)3-s + 4-s + (−0.866 − 1.5i)5-s + (0.866 − 0.5i)6-s − 8-s + (0.499 − 0.866i)9-s + (0.866 + 1.5i)10-s + (−0.866 + 0.5i)12-s + (1.5 + 0.866i)15-s + 16-s + (−0.499 + 0.866i)18-s + (−0.866 + 1.5i)19-s + (−0.866 − 1.5i)20-s + (0.5 + 0.866i)23-s + (0.866 − 0.5i)24-s + ⋯
L(s)  = 1  − 2-s + (−0.866 + 0.5i)3-s + 4-s + (−0.866 − 1.5i)5-s + (0.866 − 0.5i)6-s − 8-s + (0.499 − 0.866i)9-s + (0.866 + 1.5i)10-s + (−0.866 + 0.5i)12-s + (1.5 + 0.866i)15-s + 16-s + (−0.499 + 0.866i)18-s + (−0.866 + 1.5i)19-s + (−0.866 − 1.5i)20-s + (0.5 + 0.866i)23-s + (0.866 − 0.5i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.971 - 0.235i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.971 - 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4131995306\)
\(L(\frac12)\) \(\approx\) \(0.4131995306\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 \)
good5 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 1.73T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−8.776745447750341650865724330485, −8.186202973999386416308500215023, −7.51938480425929855530316984110, −6.60332544169185500293389564696, −5.72354796510526699632570690900, −5.12627555295153848654115432309, −4.13848183942837512099984201199, −3.50234537701979501605034463044, −1.75789353772282664738240592557, −0.792385404286588407875381672765, 0.56693079551653595504041530247, 2.19544701259194308732220256223, 2.83194160759533188908704808198, 3.96926537487367800893808822558, 5.07625010715186454669430892054, 6.24105101734465867043662240252, 6.74491987843198364344291574240, 7.09795479877242076817001845962, 7.87722573830087677518692069559, 8.518200243068385318186284660042

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