Properties

Degree $2$
Conductor $3528$
Sign $0.953 - 0.300i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)5-s + (1.22 − 0.707i)11-s + 13-s + (−0.5 + 0.866i)19-s + (0.499 + 0.866i)25-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s − 1.41i·41-s − 43-s + (−1.22 − 0.707i)47-s + 2·55-s + (1.22 + 0.707i)65-s + (0.5 + 0.866i)67-s − 1.41i·71-s + (0.5 + 0.866i)73-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)5-s + (1.22 − 0.707i)11-s + 13-s + (−0.5 + 0.866i)19-s + (0.499 + 0.866i)25-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s − 1.41i·41-s − 43-s + (−1.22 − 0.707i)47-s + 2·55-s + (1.22 + 0.707i)65-s + (0.5 + 0.866i)67-s − 1.41i·71-s + (0.5 + 0.866i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.953 - 0.300i$
Motivic weight: \(0\)
Character: $\chi_{3528} (1745, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.953 - 0.300i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.765753930\)
\(L(\frac12)\) \(\approx\) \(1.765753930\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.778838252342251208226100578777, −8.245157701645594619010576784110, −7.07269490053752020366963863634, −6.35029311160322581954581122817, −6.05464373439526974414179292244, −5.21209064831300390378905128625, −3.89622480377167573362255900124, −3.38874110237557171302395999255, −2.16898883701288896554781900893, −1.38735657387229775615645375447, 1.32584255438721064839670523976, 1.89520071877361127533978749223, 3.17671985002054231632029758597, 4.22209425775612428703850884425, 4.93073349739182697894477832075, 5.74284353428467623929114088432, 6.51109265396773838243124495833, 6.95566824828564566732306195737, 8.232265134921934373854119286456, 8.809664460173835891977841812462

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