Properties

Label 2-51480-1.1-c1-0-43
Degree $2$
Conductor $51480$
Sign $-1$
Analytic cond. $411.069$
Root an. cond. $20.2748$
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  + 5-s + 4·7-s + 11-s + 13-s − 4·17-s + 6·19-s − 2·23-s + 25-s − 6·29-s − 2·31-s + 4·35-s − 8·37-s − 2·41-s − 8·43-s + 9·49-s − 12·53-s + 55-s − 12·59-s − 14·61-s + 65-s + 16·67-s + 12·71-s + 4·73-s + 4·77-s + 16·79-s + 4·83-s − 4·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s + 0.301·11-s + 0.277·13-s − 0.970·17-s + 1.37·19-s − 0.417·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s + 0.676·35-s − 1.31·37-s − 0.312·41-s − 1.21·43-s + 9/7·49-s − 1.64·53-s + 0.134·55-s − 1.56·59-s − 1.79·61-s + 0.124·65-s + 1.95·67-s + 1.42·71-s + 0.468·73-s + 0.455·77-s + 1.80·79-s + 0.439·83-s − 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51480\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13\)
Sign: $-1$
Analytic conductor: \(411.069\)
Root analytic conductor: \(20.2748\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{51480} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 51480,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.77909119690900, −14.05439163377299, −13.77850676723721, −13.52345261742888, −12.54664135931830, −12.23419809426371, −11.54935079504562, −11.06291514256824, −10.89246522697299, −10.10004388048492, −9.393318556492147, −9.138371984829740, −8.446190273069021, −7.827417902817942, −7.592413834721200, −6.645090467405651, −6.372484662373269, −5.315854128386520, −5.204686210577887, −4.583162628784883, −3.758339912639363, −3.266827199339204, −2.214827301863700, −1.735076995017091, −1.221210341344202, 0, 1.221210341344202, 1.735076995017091, 2.214827301863700, 3.266827199339204, 3.758339912639363, 4.583162628784883, 5.204686210577887, 5.315854128386520, 6.372484662373269, 6.645090467405651, 7.592413834721200, 7.827417902817942, 8.446190273069021, 9.138371984829740, 9.393318556492147, 10.10004388048492, 10.89246522697299, 11.06291514256824, 11.54935079504562, 12.23419809426371, 12.54664135931830, 13.52345261742888, 13.77850676723721, 14.05439163377299, 14.77909119690900

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