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Properties

Label 2-3528-7.4-c1-0-11
Degree $2$
Conductor $3528$
Sign $0.991 - 0.126i$
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 − 6·13-s + (1 − 1.73i)17-s + (−2 − 3.46i)19-s + (−1 − 1.73i)23-s + (0.500 − 0.866i)25-s + 8·29-s + (−2 + 3.46i)31-s + (3 + 5.19i)37-s + 10·41-s − 4·43-s + (2 + 3.46i)47-s + (2 − 3.46i)53-s + 12·55-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s + (−0.904 + 1.56i)11-s − 1.66·13-s + (0.242 − 0.420i)17-s + (−0.458 − 0.794i)19-s + (−0.208 − 0.361i)23-s + (0.100 − 0.173i)25-s + 1.48·29-s + (−0.359 + 0.622i)31-s + (0.493 + 0.854i)37-s + 1.56·41-s − 0.609·43-s + (0.291 + 0.505i)47-s + (0.274 − 0.475i)53-s + 1.61·55-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: $0.991 - 0.126i$
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: \((2,\ 3528,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.162553584\)
\(L(\frac12)\) \(\approx\) \(1.162553584\)
\(L(\frac{3}{2})\) 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 \( 1 + 6T + 13T^{2} \)
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 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
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 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.492474415400537979106175373553, −7.81835855618017360641556395425, −7.21127570555540978533765688535, −6.54699529632811106836418218097, −5.15792271346531491338176029494, −4.81902058869356988402700580510, −4.28911323581950568590629415399, −2.77495405506953136436756080208, −2.21809125443337599213833066095, −0.68141682516951546714755150837, 0.54236527448166837249654009028, 2.25900572574214636740272990688, 2.97584687203881023110663639492, 3.74935043955816335674645271539, 4.74657430411269896256017203262, 5.68188692185819798270310781399, 6.21116920818554400341986104158, 7.30748730506948330074668685119, 7.71204472913453961375092288518, 8.403407073208189605002474165311

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