Properties

Label 2-60984-1.1-c1-0-29
Degree $2$
Conductor $60984$
Sign $-1$
Analytic cond. $486.959$
Root an. cond. $22.0671$
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 − 7-s + 13-s − 2·17-s + 19-s − 4·23-s − 4·25-s − 5·29-s − 4·31-s + 35-s − 3·37-s + 6·41-s − 2·43-s + 9·47-s + 49-s − 2·53-s + 59-s + 2·61-s − 65-s − 11·67-s + 2·71-s + 11·73-s + 14·79-s + 6·83-s + 2·85-s + 14·89-s − 91-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 0.277·13-s − 0.485·17-s + 0.229·19-s − 0.834·23-s − 4/5·25-s − 0.928·29-s − 0.718·31-s + 0.169·35-s − 0.493·37-s + 0.937·41-s − 0.304·43-s + 1.31·47-s + 1/7·49-s − 0.274·53-s + 0.130·59-s + 0.256·61-s − 0.124·65-s − 1.34·67-s + 0.237·71-s + 1.28·73-s + 1.57·79-s + 0.658·83-s + 0.216·85-s + 1.48·89-s − 0.104·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60984\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(486.959\)
Root analytic conductor: \(22.0671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{60984} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 60984,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good5 \( 1 + T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−14.67025218343958, −13.96162014997266, −13.43893988078364, −13.21224789806418, −12.28590760638222, −12.19938677817537, −11.52940056746451, −10.91877369394589, −10.63053105809791, −9.913451865040977, −9.294883045918612, −9.062992625593861, −8.274316421292874, −7.740854629989106, −7.381581378420666, −6.671737954351313, −6.114537246676636, −5.618016787835523, −4.980890783210437, −4.182603384598716, −3.775954518881538, −3.264628634656668, −2.309269899201325, −1.861250275341136, −0.7911467041342091, 0, 0.7911467041342091, 1.861250275341136, 2.309269899201325, 3.264628634656668, 3.775954518881538, 4.182603384598716, 4.980890783210437, 5.618016787835523, 6.114537246676636, 6.671737954351313, 7.381581378420666, 7.740854629989106, 8.274316421292874, 9.062992625593861, 9.294883045918612, 9.913451865040977, 10.63053105809791, 10.91877369394589, 11.52940056746451, 12.19938677817537, 12.28590760638222, 13.21224789806418, 13.43893988078364, 13.96162014997266, 14.67025218343958

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