Properties

Label 2-60984-1.1-c1-0-58
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 7-s − 7·13-s + 2·17-s + 19-s + 6·23-s + 4·25-s − 7·29-s − 10·31-s − 3·35-s − 7·37-s − 6·41-s + 8·43-s − 13·47-s + 49-s − 10·53-s − 9·59-s + 6·61-s + 21·65-s + 67-s + 6·71-s − 13·73-s − 8·79-s − 18·83-s − 6·85-s + 2·89-s − 7·91-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.377·7-s − 1.94·13-s + 0.485·17-s + 0.229·19-s + 1.25·23-s + 4/5·25-s − 1.29·29-s − 1.79·31-s − 0.507·35-s − 1.15·37-s − 0.937·41-s + 1.21·43-s − 1.89·47-s + 1/7·49-s − 1.37·53-s − 1.17·59-s + 0.768·61-s + 2.60·65-s + 0.122·67-s + 0.712·71-s − 1.52·73-s − 0.900·79-s − 1.97·83-s − 0.650·85-s + 0.211·89-s − 0.733·91-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−14.91340044105715, −14.43332116531180, −14.00359183630351, −13.03320482201835, −12.68112056154042, −12.33451244138647, −11.68761075782772, −11.32552062017393, −10.92198351398729, −10.23965752469431, −9.593104993163013, −9.219774777946471, −8.549021629794155, −7.918213859128748, −7.498851854352673, −7.170013338701291, −6.705164022841895, −5.464083087981123, −5.342085671653135, −4.606672932598485, −4.135508176738912, −3.297624753680307, −3.032510322860639, −2.014238531547959, −1.372518518792123, 0, 0, 1.372518518792123, 2.014238531547959, 3.032510322860639, 3.297624753680307, 4.135508176738912, 4.606672932598485, 5.342085671653135, 5.464083087981123, 6.705164022841895, 7.170013338701291, 7.498851854352673, 7.918213859128748, 8.549021629794155, 9.219774777946471, 9.593104993163013, 10.23965752469431, 10.92198351398729, 11.32552062017393, 11.68761075782772, 12.33451244138647, 12.68112056154042, 13.03320482201835, 14.00359183630351, 14.43332116531180, 14.91340044105715

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