Properties

Label 2-60984-1.1-c1-0-28
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 7-s − 4·13-s + 4·19-s − 4·23-s − 25-s + 10·29-s + 2·31-s + 2·35-s + 10·37-s − 4·43-s + 2·47-s + 49-s − 2·53-s − 6·59-s + 8·65-s − 8·67-s − 12·71-s + 12·73-s − 16·79-s − 4·83-s + 6·89-s + 4·91-s − 8·95-s − 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 1.10·13-s + 0.917·19-s − 0.834·23-s − 1/5·25-s + 1.85·29-s + 0.359·31-s + 0.338·35-s + 1.64·37-s − 0.609·43-s + 0.291·47-s + 1/7·49-s − 0.274·53-s − 0.781·59-s + 0.992·65-s − 0.977·67-s − 1.42·71-s + 1.40·73-s − 1.80·79-s − 0.439·83-s + 0.635·89-s + 0.419·91-s − 0.820·95-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: \(1\)
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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.57611779825218, −14.02066696791545, −13.62085825922501, −12.99451554145046, −12.29070922899755, −12.08583392942874, −11.65514536130795, −11.09949968716310, −10.38779751571107, −9.831512161508484, −9.659188267046544, −8.809235690041264, −8.284426288762944, −7.667801475264110, −7.468982865171288, −6.705873856311600, −6.189825244150290, −5.559359353826264, −4.790987258726228, −4.389797045136993, −3.819789809407822, −2.873168262368571, −2.780151762136329, −1.673122663708715, −0.7755741783368163, 0, 0.7755741783368163, 1.673122663708715, 2.780151762136329, 2.873168262368571, 3.819789809407822, 4.389797045136993, 4.790987258726228, 5.559359353826264, 6.189825244150290, 6.705873856311600, 7.468982865171288, 7.667801475264110, 8.284426288762944, 8.809235690041264, 9.659188267046544, 9.831512161508484, 10.38779751571107, 11.09949968716310, 11.65514536130795, 12.08583392942874, 12.29070922899755, 12.99451554145046, 13.62085825922501, 14.02066696791545, 14.57611779825218

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