Properties

Label 2-60984-1.1-c1-0-16
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  + 5-s + 7-s + 4·13-s − 3·17-s + 8·19-s − 4·23-s − 4·25-s + 8·29-s − 10·31-s + 35-s − 2·37-s + 6·41-s + 43-s − 13·47-s + 49-s − 8·53-s + 3·59-s + 4·65-s + 7·67-s + 6·71-s + 6·73-s − 8·79-s + 83-s − 3·85-s + 3·89-s + 4·91-s + 8·95-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.10·13-s − 0.727·17-s + 1.83·19-s − 0.834·23-s − 4/5·25-s + 1.48·29-s − 1.79·31-s + 0.169·35-s − 0.328·37-s + 0.937·41-s + 0.152·43-s − 1.89·47-s + 1/7·49-s − 1.09·53-s + 0.390·59-s + 0.496·65-s + 0.855·67-s + 0.712·71-s + 0.702·73-s − 0.900·79-s + 0.109·83-s − 0.325·85-s + 0.317·89-s + 0.419·91-s + 0.820·95-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

\(L(1)\) \(\approx\) \(2.886580508\)
\(L(\frac12)\) \(\approx\) \(2.886580508\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−14.14818057702236, −13.92140290720741, −13.33254657901481, −12.87442705541369, −12.26945022845575, −11.67436359124668, −11.18451634223508, −10.93760396579999, −10.05486899305893, −9.761934374869297, −9.173352699447449, −8.649079172307132, −8.026326753773170, −7.656706442083605, −6.910780444710249, −6.361654027562745, −5.841278184063201, −5.286835056448586, −4.764041003006345, −3.950170278270117, −3.491327275633787, −2.780208329362726, −1.913809498738170, −1.470934794746534, −0.5850999093292416, 0.5850999093292416, 1.470934794746534, 1.913809498738170, 2.780208329362726, 3.491327275633787, 3.950170278270117, 4.764041003006345, 5.286835056448586, 5.841278184063201, 6.361654027562745, 6.910780444710249, 7.656706442083605, 8.026326753773170, 8.649079172307132, 9.173352699447449, 9.761934374869297, 10.05486899305893, 10.93760396579999, 11.18451634223508, 11.67436359124668, 12.26945022845575, 12.87442705541369, 13.33254657901481, 13.92140290720741, 14.14818057702236

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