Properties

Degree $2$
Conductor $3528$
Sign $-0.487 - 0.873i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.608 + 0.793i)3-s + (0.499 + 0.866i)4-s + (0.130 + 0.991i)6-s + 0.999i·8-s + (−0.258 + 0.965i)9-s + (0.448 + 0.258i)11-s + (−0.382 + 0.923i)12-s + (−0.5 + 0.866i)16-s + 0.261·17-s + (−0.707 + 0.707i)18-s − 1.21i·19-s + (0.258 + 0.448i)22-s + (−0.793 + 0.608i)24-s + (−0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.608 + 0.793i)3-s + (0.499 + 0.866i)4-s + (0.130 + 0.991i)6-s + 0.999i·8-s + (−0.258 + 0.965i)9-s + (0.448 + 0.258i)11-s + (−0.382 + 0.923i)12-s + (−0.5 + 0.866i)16-s + 0.261·17-s + (−0.707 + 0.707i)18-s − 1.21i·19-s + (0.258 + 0.448i)22-s + (−0.793 + 0.608i)24-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.487 - 0.873i$
Motivic weight: \(0\)
Character: $\chi_{3528} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.487 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.611689641\)
\(L(\frac12)\) \(\approx\) \(2.611689641\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.608 - 0.793i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - 0.261T + T^{2} \)
19 \( 1 + 1.21iT - T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.991 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.793 - 1.37i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.58iT - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - 0.765T + T^{2} \)
97 \( 1 + (1.71 + 0.991i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.872007370762637792104384285424, −8.310053903914964569503533870551, −7.25953150681946120444819687892, −6.98761693325927716264569393635, −5.67050623167462367016245697482, −5.25731914938164314297153202761, −4.27179271735728640293692261558, −3.77368313028983901585119746049, −2.86535088179793387327823775083, −2.00121805628386355265720667484, 1.13905967885759485009529492018, 2.00345337489757620449922827930, 2.97590094077528268744378773019, 3.67091334745111315546372060916, 4.47654363730467775076506220524, 5.59021132613396098366061087132, 6.28966849207096819347532932140, 6.76882096994920508766385322087, 7.85950933914355049341142636890, 8.272896080907045106052054404684

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