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Properties

Label 2-1488-93.47-c0-0-0
Degree $2$
Conductor $1488$
Sign $0.999 - 0.0148i$
Analytic cond. $0.742608$
Root an. cond. $0.861747$
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  + (0.809 + 0.587i)3-s + (0.5 − 1.53i)7-s + (0.309 + 0.951i)9-s + (−0.5 − 0.363i)13-s + (0.5 − 0.363i)19-s + (1.30 − 0.951i)21-s + 25-s + (−0.309 + 0.951i)27-s + (0.809 + 0.587i)31-s − 1.61·37-s + (−0.190 − 0.587i)39-s + (−1.30 + 0.951i)43-s + (−1.30 − 0.951i)49-s + 0.618·57-s + 0.618·61-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + (0.5 − 1.53i)7-s + (0.309 + 0.951i)9-s + (−0.5 − 0.363i)13-s + (0.5 − 0.363i)19-s + (1.30 − 0.951i)21-s + 25-s + (−0.309 + 0.951i)27-s + (0.809 + 0.587i)31-s − 1.61·37-s + (−0.190 − 0.587i)39-s + (−1.30 + 0.951i)43-s + (−1.30 − 0.951i)49-s + 0.618·57-s + 0.618·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1488\)    =    \(2^{4} \cdot 3 \cdot 31\)
Sign: $0.999 - 0.0148i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1488} (977, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1488,\ (\ :0),\ 0.999 - 0.0148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.504582540\)
\(L(\frac12)\) \(\approx\) \(1.504582540\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
good5 \( 1 - T^{2} \)
7 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + 1.61T + T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 + 0.618T + T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.864927441176265800709225237753, −8.819043005242931912960336531212, −8.131718736525718296272021863471, −7.35836206754327133862842338329, −6.77065082541891905727893728622, −5.15462758307147226714010521129, −4.63565855609123401904022458158, −3.67843462891121149306594932282, −2.84318413770895777631909381435, −1.37873456310526249170272451904, 1.66627838901613839352309436874, 2.50354207201746235452166729523, 3.40315464620159521524429660193, 4.74039603216732057396849673626, 5.58542848718481292594954629898, 6.54684092144748555003067577882, 7.34151589119615679965503225055, 8.297015965784398800189178866524, 8.729524858801416154724448558150, 9.440902918215098949800515797820

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