Properties

Label 2-60984-1.1-c1-0-61
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  − 5-s + 7-s − 13-s − 8·17-s − 5·19-s − 4·25-s + 5·29-s − 6·31-s − 35-s − 11·37-s − 8·43-s − 7·47-s + 49-s − 2·53-s − 7·59-s + 2·61-s + 65-s − 7·67-s − 7·73-s − 2·79-s − 12·83-s + 8·85-s − 2·89-s − 91-s + 5·95-s − 8·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.277·13-s − 1.94·17-s − 1.14·19-s − 4/5·25-s + 0.928·29-s − 1.07·31-s − 0.169·35-s − 1.80·37-s − 1.21·43-s − 1.02·47-s + 1/7·49-s − 0.274·53-s − 0.911·59-s + 0.256·61-s + 0.124·65-s − 0.855·67-s − 0.819·73-s − 0.225·79-s − 1.31·83-s + 0.867·85-s − 0.211·89-s − 0.104·91-s + 0.512·95-s − 0.812·97-s + 0.0995·101-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: \(2\)
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 \( 1 + T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.99140250951945, −14.28166635704180, −13.77345474560792, −13.28653376053629, −12.79696831365142, −12.26050812728450, −11.72365162572515, −11.22847505165528, −10.83245459009197, −10.28690755053387, −9.737617319473480, −8.957840994553419, −8.591451776912765, −8.256586789609768, −7.455066529463297, −6.979055065480626, −6.492794652271243, −5.920279847112559, −5.080886417384186, −4.636019720726400, −4.140822960595635, −3.501680165823352, −2.727983685099048, −1.967464471674923, −1.561536782890750, 0, 0, 1.561536782890750, 1.967464471674923, 2.727983685099048, 3.501680165823352, 4.140822960595635, 4.636019720726400, 5.080886417384186, 5.920279847112559, 6.492794652271243, 6.979055065480626, 7.455066529463297, 8.256586789609768, 8.591451776912765, 8.957840994553419, 9.737617319473480, 10.28690755053387, 10.83245459009197, 11.22847505165528, 11.72365162572515, 12.26050812728450, 12.79696831365142, 13.28653376053629, 13.77345474560792, 14.28166635704180, 14.99140250951945

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