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Properties

Label 2-3528-504.419-c0-0-7
Degree $2$
Conductor $3528$
Sign $-0.487 + 0.873i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
Motivic weight $0$
Arithmetic yes
Rational no
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: $-0.487 + 0.873i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2939, \cdot )$
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

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

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