Properties

Degree $2$
Conductor $3528$
Sign $0.0348 - 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0348 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0348 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.0348 - 0.999i$
Motivic weight: \(0\)
Character: $\chi_{3528} (1745, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.0348 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5762696665\)
\(L(\frac12)\) \(\approx\) \(0.5762696665\)
\(L(1)\) 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

−8.611570057777741821033017916687, −8.196933830183564652033934561007, −7.68297674420443987053660628460, −6.83045016045643918950848124808, −5.84169771051646778767065579917, −5.05481438898857624395573915695, −4.26081058188177777243893952061, −3.67350046840134224942105899317, −2.54349786270367631372483893525, −1.26395969672897782674459178465, 0.35933448142704463429978873713, 2.15193637122187343174648184778, 3.26495586219787160874483694853, 3.64116853052654115677327253668, 4.71808483062269148402882622766, 5.56070801828993697873976343942, 6.43320689727932498710165543849, 7.24549952664142577611807395226, 7.73431400371598179443485331954, 8.606882260665573496659626337379

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