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Properties

Label 2-3528-392.125-c0-0-1
Degree $2$
Conductor $3528$
Sign $-0.926 + 0.375i$
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  + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.193 + 0.846i)5-s + (−0.222 − 0.974i)7-s + (0.974 + 0.222i)8-s + (0.846 − 0.193i)10-s + (−0.541 − 1.12i)11-s + (−0.781 + 0.623i)14-s + (−0.222 − 0.974i)16-s + (−0.541 − 0.678i)20-s + (−0.777 + 0.974i)22-s + (0.222 + 0.107i)25-s + (0.900 + 0.433i)28-s + (−1.40 + 1.12i)29-s − 1.94i·31-s + (−0.781 + 0.623i)32-s + ⋯
L(s)  = 1  + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.193 + 0.846i)5-s + (−0.222 − 0.974i)7-s + (0.974 + 0.222i)8-s + (0.846 − 0.193i)10-s + (−0.541 − 1.12i)11-s + (−0.781 + 0.623i)14-s + (−0.222 − 0.974i)16-s + (−0.541 − 0.678i)20-s + (−0.777 + 0.974i)22-s + (0.222 + 0.107i)25-s + (0.900 + 0.433i)28-s + (−1.40 + 1.12i)29-s − 1.94i·31-s + (−0.781 + 0.623i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.926 + 0.375i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.926 + 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5237411522\)
\(L(\frac12)\) \(\approx\) \(0.5237411522\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.433 + 0.900i)T \)
3 \( 1 \)
7 \( 1 + (0.222 + 0.974i)T \)
good5 \( 1 + (0.193 - 0.846i)T + (-0.900 - 0.433i)T^{2} \)
11 \( 1 + (0.541 + 1.12i)T + (-0.623 + 0.781i)T^{2} \)
13 \( 1 + (0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.222 - 0.974i)T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.222 - 0.974i)T^{2} \)
29 \( 1 + (1.40 - 1.12i)T + (0.222 - 0.974i)T^{2} \)
31 \( 1 + 1.94iT - T^{2} \)
37 \( 1 + (0.222 - 0.974i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (1.56 + 1.24i)T + (0.222 + 0.974i)T^{2} \)
59 \( 1 + (-0.900 + 0.433i)T^{2} \)
61 \( 1 + (-0.222 + 0.974i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.846 + 1.75i)T + (-0.623 - 0.781i)T^{2} \)
79 \( 1 - 0.445T + T^{2} \)
83 \( 1 + (0.781 + 0.376i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.623 - 0.781i)T^{2} \)
97 \( 1 + 1.56iT - T^{2} \)
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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

−8.428729178815066064449581762677, −7.72557793410280489898768823657, −7.22165983791125901653257893948, −6.30827236605906055761699402620, −5.29621488670419910542355992327, −4.24270961024563044687391819556, −3.43652776871820869854413659630, −2.99667303609928654254834708226, −1.78241735671873430710706675628, −0.36381774646015280213237765448, 1.42039951652485978987595773558, 2.49303452135074747106831786691, 3.92204073922188181457301479282, 4.91177491232108791511106459847, 5.22050120971158939855330931131, 6.13389681129448791462161497406, 6.90120851012517025191995157118, 7.72756654599004711063607767682, 8.281256073552553887264695255755, 9.053068240229317495774437920123

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