Properties

Label 2-60984-1.1-c1-0-17
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 7-s − 2·13-s − 2·17-s + 4·19-s + 8·23-s − 25-s − 2·29-s − 4·31-s − 2·35-s + 6·37-s + 6·41-s + 4·43-s + 12·47-s + 49-s + 10·53-s − 8·59-s − 10·61-s − 4·65-s + 4·67-s + 8·71-s + 2·73-s + 8·79-s + 12·83-s − 4·85-s − 10·89-s + 2·91-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.371·29-s − 0.718·31-s − 0.338·35-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 1.37·53-s − 1.04·59-s − 1.28·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + 0.234·73-s + 0.900·79-s + 1.31·83-s − 0.433·85-s − 1.05·89-s + 0.209·91-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: \(0\)
Selberg data: \((2,\ 60984,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.944460667\)
\(L(\frac12)\) \(\approx\) \(2.944460667\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−14.08201453835262, −13.87277329489516, −13.28519830287800, −12.82174241452396, −12.41549272023897, −11.78937296700435, −11.12421164580172, −10.76455058132129, −10.20322755601525, −9.483609212003478, −9.259190073865850, −8.942330782220712, −7.973036965190888, −7.382435815776008, −7.099869675445421, −6.294917084560235, −5.856829997103700, −5.309531887783272, −4.785456659225920, −4.036920094741975, −3.375964675130138, −2.571576588580641, −2.286704074099409, −1.286519370658432, −0.6158024708816638, 0.6158024708816638, 1.286519370658432, 2.286704074099409, 2.571576588580641, 3.375964675130138, 4.036920094741975, 4.785456659225920, 5.309531887783272, 5.856829997103700, 6.294917084560235, 7.099869675445421, 7.382435815776008, 7.973036965190888, 8.942330782220712, 9.259190073865850, 9.483609212003478, 10.20322755601525, 10.76455058132129, 11.12421164580172, 11.78937296700435, 12.41549272023897, 12.82174241452396, 13.28519830287800, 13.87277329489516, 14.08201453835262

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