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Properties

Label 2-3528-504.499-c0-0-0
Degree $2$
Conductor $3528$
Sign $-0.101 - 0.994i$
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 + (0.258 + 0.965i)3-s + 4-s + (−0.258 − 0.965i)6-s − 8-s + (−0.866 + 0.499i)9-s + (0.866 − 1.5i)11-s + (0.258 + 0.965i)12-s + 16-s + (0.258 + 0.448i)17-s + (0.866 − 0.499i)18-s + (−0.965 + 1.67i)19-s + (−0.866 + 1.5i)22-s + (−0.258 − 0.965i)24-s + (−0.5 + 0.866i)25-s + ⋯
L(s)  = 1  − 2-s + (0.258 + 0.965i)3-s + 4-s + (−0.258 − 0.965i)6-s − 8-s + (−0.866 + 0.499i)9-s + (0.866 − 1.5i)11-s + (0.258 + 0.965i)12-s + 16-s + (0.258 + 0.448i)17-s + (0.866 − 0.499i)18-s + (−0.965 + 1.67i)19-s + (−0.866 + 1.5i)22-s + (−0.258 − 0.965i)24-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.101 - 0.994i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.101 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8153320309\)
\(L(\frac12)\) \(\approx\) \(0.8153320309\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 - 1.93T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.944132317344535519252073867138, −8.220102090399050546151397885114, −8.014895320046813903978383815029, −6.68181689351019723257047854253, −5.98029052438660957275453582202, −5.45100387762232228481791581633, −3.91645840489377016453858614372, −3.58909442517183113588827181510, −2.49428180504702387687091645027, −1.28410901709933757559204029560, 0.68000241757530248369224877985, 2.02608884048610957420317006608, 2.38093272868110305524609405511, 3.66073822459700215429197378672, 4.78495496817376795849510816840, 5.93291105318194242892255774979, 6.79594070676162975807095744613, 7.03444448621452185575869406714, 7.75835468218964166530572185146, 8.665940221363220881986857319038

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