Properties

Label 2-60984-1.1-c1-0-48
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 + 3·13-s + 4·17-s − 19-s − 2·23-s − 4·25-s − 29-s + 4·31-s − 35-s − 3·37-s + 4·41-s − 2·43-s − 5·47-s + 49-s − 2·53-s − 9·59-s + 2·61-s + 3·65-s − 3·67-s − 12·71-s − 7·73-s + 8·79-s + 12·83-s + 4·85-s + 2·89-s − 3·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.832·13-s + 0.970·17-s − 0.229·19-s − 0.417·23-s − 4/5·25-s − 0.185·29-s + 0.718·31-s − 0.169·35-s − 0.493·37-s + 0.624·41-s − 0.304·43-s − 0.729·47-s + 1/7·49-s − 0.274·53-s − 1.17·59-s + 0.256·61-s + 0.372·65-s − 0.366·67-s − 1.42·71-s − 0.819·73-s + 0.900·79-s + 1.31·83-s + 0.433·85-s + 0.211·89-s − 0.314·91-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 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−14.41269386285797, −14.00268793882073, −13.58500080503970, −13.03381871896125, −12.64886064395590, −11.89407242184167, −11.67798896073408, −10.93409748573364, −10.31071739041389, −10.08135047068187, −9.416101938509087, −8.959590838695040, −8.367460587753938, −7.754509802536442, −7.379947323365164, −6.452381845534754, −6.154591764242589, −5.715106107879477, −4.975234547246730, −4.372313308261250, −3.575400823580493, −3.254099685168930, −2.380611911323824, −1.696850997409341, −1.015376080209245, 0, 1.015376080209245, 1.696850997409341, 2.380611911323824, 3.254099685168930, 3.575400823580493, 4.372313308261250, 4.975234547246730, 5.715106107879477, 6.154591764242589, 6.452381845534754, 7.379947323365164, 7.754509802536442, 8.367460587753938, 8.959590838695040, 9.416101938509087, 10.08135047068187, 10.31071739041389, 10.93409748573364, 11.67798896073408, 11.89407242184167, 12.64886064395590, 13.03381871896125, 13.58500080503970, 14.00268793882073, 14.41269386285797

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