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Properties

Label 2-3528-392.181-c0-0-0
Degree $2$
Conductor $3528$
Sign $-0.801 - 0.598i$
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  + (0.974 + 0.222i)2-s + (0.900 + 0.433i)4-s + (−1.21 + 1.52i)5-s + (0.623 + 0.781i)7-s + (0.781 + 0.623i)8-s + (−1.52 + 1.21i)10-s + (−1.75 − 0.400i)11-s + (0.433 + 0.900i)14-s + (0.623 + 0.781i)16-s + (−1.75 + 0.846i)20-s + (−1.62 − 0.781i)22-s + (−0.623 − 2.73i)25-s + (0.222 + 0.974i)28-s + (0.193 + 0.400i)29-s + 1.56i·31-s + (0.433 + 0.900i)32-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)2-s + (0.900 + 0.433i)4-s + (−1.21 + 1.52i)5-s + (0.623 + 0.781i)7-s + (0.781 + 0.623i)8-s + (−1.52 + 1.21i)10-s + (−1.75 − 0.400i)11-s + (0.433 + 0.900i)14-s + (0.623 + 0.781i)16-s + (−1.75 + 0.846i)20-s + (−1.62 − 0.781i)22-s + (−0.623 − 2.73i)25-s + (0.222 + 0.974i)28-s + (0.193 + 0.400i)29-s + 1.56i·31-s + (0.433 + 0.900i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.801 - 0.598i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.801 - 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.639281193\)
\(L(\frac12)\) \(\approx\) \(1.639281193\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.974 - 0.222i)T \)
3 \( 1 \)
7 \( 1 + (-0.623 - 0.781i)T \)
good5 \( 1 + (1.21 - 1.52i)T + (-0.222 - 0.974i)T^{2} \)
11 \( 1 + (1.75 + 0.400i)T + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + (-0.623 + 0.781i)T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.623 + 0.781i)T^{2} \)
29 \( 1 + (-0.193 - 0.400i)T + (-0.623 + 0.781i)T^{2} \)
31 \( 1 - 1.56iT - T^{2} \)
37 \( 1 + (-0.623 + 0.781i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.867 + 1.80i)T + (-0.623 - 0.781i)T^{2} \)
59 \( 1 + (-0.222 + 0.974i)T^{2} \)
61 \( 1 + (0.623 - 0.781i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (1.52 - 0.347i)T + (0.900 - 0.433i)T^{2} \)
79 \( 1 + 1.24T + T^{2} \)
83 \( 1 + (-0.433 - 1.90i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.900 - 0.433i)T^{2} \)
97 \( 1 + 0.867iT - T^{2} \)
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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

−8.466593101307523059145548339577, −8.209813135611692996843487880448, −7.36811949592146553396600773627, −6.93767485109281492492446624966, −5.97472294751400325609452434875, −5.24643110563883796407324070145, −4.50946997636772219207422460096, −3.41845386783042344183616901959, −2.92603069862642046869870771275, −2.19426791318614009398035654749, 0.65859681593581168766497135904, 1.86889279934522749827402414518, 3.05675601905104295932368779449, 4.23460570578047426162299630324, 4.41256566864370951046509818826, 5.19241911804340169600913195110, 5.81972468316825155403286394570, 7.26774188649334921693745379736, 7.67737797481975169297794942663, 8.122596018082269493796123475524

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