Properties

Label 2-48510-1.1-c1-0-36
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 $0$

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 + 2·17-s − 4·19-s − 20-s − 22-s + 4·23-s + 25-s + 2·26-s + 6·29-s − 32-s − 2·34-s + 2·37-s + 4·38-s + 40-s + 6·41-s + 12·43-s + 44-s − 4·46-s + 8·47-s − 50-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 + 0.485·17-s − 0.917·19-s − 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s + 1.11·29-s − 0.176·32-s − 0.342·34-s + 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s + 1.82·43-s + 0.150·44-s − 0.589·46-s + 1.16·47-s − 0.141·50-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: \(0\)
Selberg data: \((2,\ 48510,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.778339421\)
\(L(\frac12)\) \(\approx\) \(1.778339421\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.66732858067037, −14.25223492842685, −13.50723258554694, −12.85922791428212, −12.38034125040386, −12.02603861465864, −11.33169057115188, −10.94932574323759, −10.33401248664860, −9.956948754840582, −9.173743649879669, −8.891349036660239, −8.245277410491918, −7.748969713554426, −7.141330252023590, −6.778452821929981, −6.029928246559551, −5.486011285697012, −4.695128308965675, −4.118739741156643, −3.496695524594288, −2.544089859294263, −2.302562146519531, −1.046672362957583, −0.6455843197834562, 0.6455843197834562, 1.046672362957583, 2.302562146519531, 2.544089859294263, 3.496695524594288, 4.118739741156643, 4.695128308965675, 5.486011285697012, 6.029928246559551, 6.778452821929981, 7.141330252023590, 7.748969713554426, 8.245277410491918, 8.891349036660239, 9.173743649879669, 9.956948754840582, 10.33401248664860, 10.94932574323759, 11.33169057115188, 12.02603861465864, 12.38034125040386, 12.85922791428212, 13.50723258554694, 14.25223492842685, 14.66732858067037

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