Properties

Label 2-60984-1.1-c1-0-45
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 − 4·13-s + 6·17-s − 2·19-s − 2·23-s − 25-s + 2·29-s + 8·31-s − 2·35-s − 2·37-s − 6·41-s + 2·43-s + 10·47-s + 49-s + 14·53-s − 12·61-s − 8·65-s − 8·67-s − 6·71-s − 6·73-s − 4·79-s + 4·83-s + 12·85-s + 12·89-s + 4·91-s − 4·95-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 1.10·13-s + 1.45·17-s − 0.458·19-s − 0.417·23-s − 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.937·41-s + 0.304·43-s + 1.45·47-s + 1/7·49-s + 1.92·53-s − 1.53·61-s − 0.992·65-s − 0.977·67-s − 0.712·71-s − 0.702·73-s − 0.450·79-s + 0.439·83-s + 1.30·85-s + 1.27·89-s + 0.419·91-s − 0.410·95-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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.59732804451919, −13.96859776513216, −13.47352959611586, −13.28746690679808, −12.29133142014852, −12.04073780633223, −11.90005771763387, −10.69784503813172, −10.37669891529476, −9.969844641694551, −9.582336031944332, −8.939582533830635, −8.433090618857057, −7.615778644641170, −7.407798778067212, −6.540668286422234, −6.150445349763756, −5.533179993545810, −5.124854845061500, −4.353761788606898, −3.777582109430398, −2.845621010236598, −2.573200906671915, −1.728671078560310, −1.006333823922483, 0, 1.006333823922483, 1.728671078560310, 2.573200906671915, 2.845621010236598, 3.777582109430398, 4.353761788606898, 5.124854845061500, 5.533179993545810, 6.150445349763756, 6.540668286422234, 7.407798778067212, 7.615778644641170, 8.433090618857057, 8.939582533830635, 9.582336031944332, 9.969844641694551, 10.37669891529476, 10.69784503813172, 11.90005771763387, 12.04073780633223, 12.29133142014852, 13.28746690679808, 13.47352959611586, 13.96859776513216, 14.59732804451919

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