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Properties

Label 2-3528-21.20-c1-0-29
Degree $2$
Conductor $3528$
Sign $0.239 + 0.970i$
Analytic cond. $28.1712$
Root an. cond. $5.30765$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.17·5-s + 1.67i·11-s − 1.14i·13-s − 4.15·17-s + 2.11i·19-s − 4.87i·23-s − 3.62·25-s − 8.32i·29-s − 8.29i·31-s + 4.39·37-s + 2.67·41-s + 4.08·43-s − 3.50·47-s − 1.84i·53-s + 1.96i·55-s + ⋯
L(s)  = 1  + 0.525·5-s + 0.503i·11-s − 0.317i·13-s − 1.00·17-s + 0.485i·19-s − 1.01i·23-s − 0.724·25-s − 1.54i·29-s − 1.48i·31-s + 0.723·37-s + 0.417·41-s + 0.623·43-s − 0.511·47-s − 0.253i·53-s + 0.264i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.239 + 0.970i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ 0.239 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.563576958\)
\(L(\frac12)\) \(\approx\) \(1.563576958\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.17T + 5T^{2} \)
11 \( 1 - 1.67iT - 11T^{2} \)
13 \( 1 + 1.14iT - 13T^{2} \)
17 \( 1 + 4.15T + 17T^{2} \)
19 \( 1 - 2.11iT - 19T^{2} \)
23 \( 1 + 4.87iT - 23T^{2} \)
29 \( 1 + 8.32iT - 29T^{2} \)
31 \( 1 + 8.29iT - 31T^{2} \)
37 \( 1 - 4.39T + 37T^{2} \)
41 \( 1 - 2.67T + 41T^{2} \)
43 \( 1 - 4.08T + 43T^{2} \)
47 \( 1 + 3.50T + 47T^{2} \)
53 \( 1 + 1.84iT - 53T^{2} \)
59 \( 1 - 5.97T + 59T^{2} \)
61 \( 1 + 14.7iT - 61T^{2} \)
67 \( 1 - 8.44T + 67T^{2} \)
71 \( 1 - 5.48iT - 71T^{2} \)
73 \( 1 + 0.977iT - 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 - 9.52iT - 97T^{2} \)
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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

−8.204425406829052139996862016919, −7.889084613095273307733685333675, −6.78665589535333474377718898334, −6.19793841787995780013125663204, −5.50005992162982673065725210955, −4.47742541893242121433399807336, −3.91171190691530091652649297163, −2.52835213049058717051496996821, −2.01767533270470047480682713549, −0.47790571967599321996553789790, 1.18935179914575777022301839601, 2.21493410992090504346773249378, 3.17019774794120167571271725373, 4.09018727295220708853376183868, 5.03851960391383442700091825473, 5.68892649604562559687119569701, 6.55810121033854742494789819435, 7.11924044869359041272081641936, 8.033900781453314103135900260212, 8.921699718658272088935863727984

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