Properties

Label 2-60984-1.1-c1-0-60
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 − 3·13-s − 7·19-s − 6·23-s − 4·25-s − 9·29-s − 35-s − 3·37-s + 8·41-s − 10·43-s − 3·47-s + 49-s − 6·53-s − 7·59-s − 10·61-s + 3·65-s − 3·67-s + 8·71-s + 7·73-s − 8·79-s + 6·89-s − 3·91-s + 7·95-s − 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.832·13-s − 1.60·19-s − 1.25·23-s − 4/5·25-s − 1.67·29-s − 0.169·35-s − 0.493·37-s + 1.24·41-s − 1.52·43-s − 0.437·47-s + 1/7·49-s − 0.824·53-s − 0.911·59-s − 1.28·61-s + 0.372·65-s − 0.366·67-s + 0.949·71-s + 0.819·73-s − 0.900·79-s + 0.635·89-s − 0.314·91-s + 0.718·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 + 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.83266359001661, −14.39628461008312, −13.78819582944208, −13.32621505534830, −12.59113857620353, −12.37435140946439, −11.78341270800800, −11.19988341760832, −10.86082421099586, −10.21543266407744, −9.664424769185181, −9.229978168648298, −8.492849148757135, −8.029137011878854, −7.625453825084384, −7.089162174677807, −6.288532192690854, −5.973581919357300, −5.150333225013615, −4.648776146410089, −3.997532297436442, −3.630044006268864, −2.664573373715894, −2.016939039021853, −1.538513719866807, 0, 0, 1.538513719866807, 2.016939039021853, 2.664573373715894, 3.630044006268864, 3.997532297436442, 4.648776146410089, 5.150333225013615, 5.973581919357300, 6.288532192690854, 7.089162174677807, 7.625453825084384, 8.029137011878854, 8.492849148757135, 9.229978168648298, 9.664424769185181, 10.21543266407744, 10.86082421099586, 11.19988341760832, 11.78341270800800, 12.37435140946439, 12.59113857620353, 13.32621505534830, 13.78819582944208, 14.39628461008312, 14.83266359001661

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