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Properties

Label 2-3528-392.13-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: $\chi_{3528} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.801 - 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.260377219\)
\(L(\frac12)\) \(\approx\) \(1.260377219\)
\(L(1)\) 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

−8.974486592849155985263980388671, −8.032666282905063224035685675657, −7.20029404655801471029512304192, −6.71635205019667867582092688660, −6.17982671465008754594043294811, −5.43882239628384785856273092777, −4.00511270439899210669671384771, −3.11417560737545487677894928610, −2.05243625309426779492879982118, −1.33222459988885924544508076771, 1.39841160960837897124837265435, 1.56868568494767259574494758096, 2.71567722832854322871880998017, 4.13788454896615090758981685855, 4.91858883333007557529768746998, 5.87075129836627568852603045953, 6.33488339901457544906363977072, 7.34138673299774957527008985528, 8.358188174665780110627333046957, 8.890406974298605137618473069168

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