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Properties

Label 2-3528-7.4-c1-0-25
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  + (−1 − 1.73i)5-s + (−3 + 5.19i)11-s + 3·13-s + (−2 + 3.46i)17-s + (−2.5 − 4.33i)19-s + (−2 − 3.46i)23-s + (0.500 − 0.866i)25-s + 4·29-s + (3.5 − 6.06i)31-s + (4.5 + 7.79i)37-s − 2·41-s − 43-s + (−1 − 1.73i)47-s + (4 − 6.92i)53-s + 12·55-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s + (−0.904 + 1.56i)11-s + 0.832·13-s + (−0.485 + 0.840i)17-s + (−0.573 − 0.993i)19-s + (−0.417 − 0.722i)23-s + (0.100 − 0.173i)25-s + 0.742·29-s + (0.628 − 1.08i)31-s + (0.739 + 1.28i)37-s − 0.312·41-s − 0.152·43-s + (−0.145 − 0.252i)47-s + (0.549 − 0.951i)53-s + 1.61·55-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

\(L(1)\) \(\approx\) \(1.363539781\)
\(L(\frac12)\) \(\approx\) \(1.363539781\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4 + 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 12.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-4 - 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−8.177142072212803551387218772635, −8.133500421065872882683525161474, −6.87023499626272756386024687317, −6.39171474401558042544198356079, −5.25422803076883456198732105243, −4.48781435053853499175980300544, −4.15624859052824832937202198813, −2.73014992314598800075977077435, −1.89682433560961563589992382436, −0.53734962439645679382839310971, 0.864726339517738493634694224314, 2.36524110087633703527091492323, 3.25644039027928428083143820484, 3.76292435624383440991346307926, 4.93758288831640695408973174555, 5.82086716386566217178480872908, 6.37081219292544425564948666557, 7.25071342869270702159529916655, 8.014107016678255440021689612767, 8.513785836516523715034270415574

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