Properties

Degree $2$
Conductor $3528$
Sign $-0.198 + 0.980i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 0.999·8-s + (−0.5 − 0.866i)16-s + (0.707 − 1.22i)17-s + (−0.707 − 1.22i)19-s + (−0.5 + 0.866i)25-s + (−0.499 + 0.866i)32-s − 1.41·34-s + (−0.707 + 1.22i)38-s + 1.41·41-s + 0.999·50-s + (0.707 − 1.22i)59-s + 0.999·64-s + (1 − 1.73i)67-s + (0.707 + 1.22i)68-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 0.999·8-s + (−0.5 − 0.866i)16-s + (0.707 − 1.22i)17-s + (−0.707 − 1.22i)19-s + (−0.5 + 0.866i)25-s + (−0.499 + 0.866i)32-s − 1.41·34-s + (−0.707 + 1.22i)38-s + 1.41·41-s + 0.999·50-s + (0.707 − 1.22i)59-s + 0.999·64-s + (1 − 1.73i)67-s + (0.707 + 1.22i)68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.198 + 0.980i$
Motivic weight: \(0\)
Character: $\chi_{3528} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.198 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8554519881\)
\(L(\frac12)\) \(\approx\) \(0.8554519881\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−8.786730798574142798841957239021, −7.82263948952795508381687594057, −7.34561667366881049636912336533, −6.45279465060330201168052791981, −5.27352421058608895990154443443, −4.63522485560747612613519669813, −3.65534635771493383146997571848, −2.85220503842300141855344897917, −1.99380360492751140790703285030, −0.67591441734767280284797769524, 1.21473151227846289722687204000, 2.29823547647708984712685316879, 3.82458300370464019627451903443, 4.34608589508292863984545091000, 5.60627014802211170006788264246, 5.92067638445988745052820920810, 6.74912973938438901991341209043, 7.59161222418283715977337260793, 8.259598995173777107442554087910, 8.640323968566846055052344707089

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