Properties

Degree $2$
Conductor $3528$
Sign $0.991 + 0.126i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)5-s + 6·13-s + (−1 − 1.73i)17-s + (−2 + 3.46i)19-s + (−2 + 3.46i)23-s + (0.500 + 0.866i)25-s + 10·29-s + (4 + 6.92i)31-s + (−3 + 5.19i)37-s + 2·41-s − 4·43-s + (4 − 6.92i)47-s + (−5 − 8.66i)53-s + (6 + 10.3i)59-s + (1 − 1.73i)61-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + 1.66·13-s + (−0.242 − 0.420i)17-s + (−0.458 + 0.794i)19-s + (−0.417 + 0.722i)23-s + (0.100 + 0.173i)25-s + 1.85·29-s + (0.718 + 1.24i)31-s + (−0.493 + 0.854i)37-s + 0.312·41-s − 0.609·43-s + (0.583 − 1.01i)47-s + (−0.686 − 1.18i)53-s + (0.781 + 1.35i)59-s + (0.128 − 0.221i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Motivic weight: \(1\)
Character: $\chi_{3528} (3313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.264443186\)
\(L(\frac12)\) \(\approx\) \(2.264443186\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5 + 8.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−8.424920741814522036659944694045, −8.218986338123309159730323495793, −6.87736678257361120134973727372, −6.34310986327447485074712883525, −5.50272507745461089850190269729, −4.83657720312791131093659864617, −3.90004202929853511830446563241, −3.07618351361672951769536253124, −1.75605530083727811500754861500, −1.00277133427536567828942054591, 0.872398733712217342574418147342, 2.18618027198153737951585901659, 2.91546801257104559277777596806, 3.94983628381681533987193806165, 4.64780565590878982475590182341, 5.88001750581901972782425508733, 6.36788753780212138931747754294, 6.83372796223109581336696846951, 8.022155646319705380616939473410, 8.502822641816571771634306320565

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