Properties

Label 2-3528-7.4-c1-0-7
Degree $2$
Conductor $3528$
Sign $-0.605 - 0.795i$
Analytic cond. $28.1712$
Root an. cond. $5.30765$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)11-s − 2·13-s + (−1.5 + 2.59i)17-s + (2.5 + 4.33i)19-s + (−1.5 − 2.59i)23-s + (2 − 3.46i)25-s + 6·29-s + (−0.5 + 0.866i)31-s + (2.5 + 4.33i)37-s − 10·41-s − 4·43-s + (−0.5 − 0.866i)47-s + (−4.5 + 7.79i)53-s − 0.999·55-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.150 + 0.261i)11-s − 0.554·13-s + (−0.363 + 0.630i)17-s + (0.573 + 0.993i)19-s + (−0.312 − 0.541i)23-s + (0.400 − 0.692i)25-s + 1.11·29-s + (−0.0898 + 0.155i)31-s + (0.410 + 0.711i)37-s − 1.56·41-s − 0.609·43-s + (−0.0729 − 0.126i)47-s + (−0.618 + 1.07i)53-s − 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.110763939\)
\(L(\frac12)\) \(\approx\) \(1.110763939\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−8.677264777718067008888600378818, −8.166423310383080954810556587018, −7.31726766036183524478256079266, −6.56202825699481683822786721348, −5.97523649418232971322541172816, −4.97960367384078813936593535257, −4.29600244192821766558779229927, −3.23116615740893548146609606076, −2.43213559524387488486262845474, −1.36663520754116046331529596047, 0.32665185366113103000616308488, 1.60678278311467160890563830409, 2.71244141787279812490716823366, 3.48890427283877296543872776650, 4.79613761986216462137133710896, 5.03798883183762878239337995928, 6.05568213534880117769942171714, 6.90607958541045003457299432342, 7.50869874515317231007050691188, 8.384801328355182377862302508174

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