Properties

Label 2-60984-1.1-c1-0-38
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  + 7-s − 2·17-s − 4·19-s − 4·23-s − 5·25-s + 6·29-s + 4·31-s + 6·37-s − 6·41-s − 4·43-s − 12·47-s + 49-s + 4·53-s + 8·61-s + 12·67-s + 4·71-s − 4·73-s + 4·83-s + 12·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·119-s + ⋯
L(s)  = 1  + 0.377·7-s − 0.485·17-s − 0.917·19-s − 0.834·23-s − 25-s + 1.11·29-s + 0.718·31-s + 0.986·37-s − 0.937·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.549·53-s + 1.02·61-s + 1.46·67-s + 0.474·71-s − 0.468·73-s + 0.439·83-s + 1.27·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.183·119-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 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.63916320984897, −13.96866608687381, −13.57440060796830, −13.09255845132126, −12.53700719251514, −11.90820643595722, −11.57205999193530, −11.06704954823045, −10.42133529830218, −9.915397984258024, −9.604585531806914, −8.725417054004016, −8.167242687759772, −8.138679059036627, −7.217025789334415, −6.537950264556073, −6.306031732559428, −5.531117263327693, −4.882894653912617, −4.387969481402981, −3.825651143297415, −3.104843895174687, −2.259820255763959, −1.884626462494536, −0.9055025743709319, 0, 0.9055025743709319, 1.884626462494536, 2.259820255763959, 3.104843895174687, 3.825651143297415, 4.387969481402981, 4.882894653912617, 5.531117263327693, 6.306031732559428, 6.537950264556073, 7.217025789334415, 8.138679059036627, 8.167242687759772, 8.725417054004016, 9.604585531806914, 9.915397984258024, 10.42133529830218, 11.06704954823045, 11.57205999193530, 11.90820643595722, 12.53700719251514, 13.09255845132126, 13.57440060796830, 13.96866608687381, 14.63916320984897

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