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Properties

Label 2-48510-1.1-c1-0-70
Degree $2$
Conductor $48510$
Sign $-1$
Analytic cond. $387.354$
Root an. cond. $19.6813$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 8-s + 10-s − 11-s + 2·13-s + 16-s + 4·17-s − 4·19-s − 20-s + 22-s + 6·23-s + 25-s − 2·26-s + 4·29-s − 8·31-s − 32-s − 4·34-s + 8·37-s + 4·38-s + 40-s − 2·41-s − 4·43-s − 44-s − 6·46-s + 8·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.554·13-s + 1/4·16-s + 0.970·17-s − 0.917·19-s − 0.223·20-s + 0.213·22-s + 1.25·23-s + 1/5·25-s − 0.392·26-s + 0.742·29-s − 1.43·31-s − 0.176·32-s − 0.685·34-s + 1.31·37-s + 0.648·38-s + 0.158·40-s − 0.312·41-s − 0.609·43-s − 0.150·44-s − 0.884·46-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48510\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(387.354\)
Root analytic conductor: \(19.6813\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 48510,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.93156013747470, −14.52434226012983, −13.70457805623882, −13.28405346294235, −12.57157160084842, −12.29716168727659, −11.64643782568614, −10.97091769164769, −10.76543885123471, −10.24138198042920, −9.446148625941234, −9.132275013647790, −8.392250982275121, −8.149869957662394, −7.340942270384861, −7.111411576096786, −6.277344868444032, −5.807546846232735, −5.121554411350576, −4.417436675275726, −3.739487727318120, −3.074445199553255, −2.512484502435011, −1.555276807023308, −0.9305846588110639, 0, 0.9305846588110639, 1.555276807023308, 2.512484502435011, 3.074445199553255, 3.739487727318120, 4.417436675275726, 5.121554411350576, 5.807546846232735, 6.277344868444032, 7.111411576096786, 7.340942270384861, 8.149869957662394, 8.392250982275121, 9.132275013647790, 9.446148625941234, 10.24138198042920, 10.76543885123471, 10.97091769164769, 11.64643782568614, 12.29716168727659, 12.57157160084842, 13.28405346294235, 13.70457805623882, 14.52434226012983, 14.93156013747470

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