Properties

Label 2-60984-1.1-c1-0-20
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  − 2·5-s − 7-s − 6·13-s − 2·17-s − 4·19-s + 4·23-s − 25-s − 10·29-s − 8·31-s + 2·35-s + 6·37-s − 2·41-s + 4·43-s − 8·47-s + 49-s + 10·53-s − 12·59-s + 2·61-s + 12·65-s + 12·67-s + 12·71-s + 14·73-s + 8·79-s + 12·83-s + 4·85-s + 2·89-s + 6·91-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 1.66·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.85·29-s − 1.43·31-s + 0.338·35-s + 0.986·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 1.37·53-s − 1.56·59-s + 0.256·61-s + 1.48·65-s + 1.46·67-s + 1.42·71-s + 1.63·73-s + 0.900·79-s + 1.31·83-s + 0.433·85-s + 0.211·89-s + 0.628·91-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
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

−14.72451440591808, −14.19032132946012, −13.32297324328218, −13.01462699129387, −12.48359462066971, −12.14752434311868, −11.42977967061912, −11.02905992639923, −10.66273295132507, −9.801648963428519, −9.369438000859030, −9.083099922609597, −8.189554839039394, −7.752868392223162, −7.324222108672214, −6.782994769144465, −6.258963083623540, −5.314701835329659, −5.076495631455014, −4.218580141436744, −3.819497958267156, −3.199714628960057, −2.288956307489984, −1.985658281789660, −0.6264667177117931, 0, 0.6264667177117931, 1.985658281789660, 2.288956307489984, 3.199714628960057, 3.819497958267156, 4.218580141436744, 5.076495631455014, 5.314701835329659, 6.258963083623540, 6.782994769144465, 7.324222108672214, 7.752868392223162, 8.189554839039394, 9.083099922609597, 9.369438000859030, 9.801648963428519, 10.66273295132507, 11.02905992639923, 11.42977967061912, 12.14752434311868, 12.48359462066971, 13.01462699129387, 13.32297324328218, 14.19032132946012, 14.72451440591808

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