Properties

Label 2-60984-1.1-c1-0-23
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  − 3·5-s − 7-s + 13-s − 5·17-s − 4·19-s + 4·23-s + 4·25-s + 7·29-s − 4·31-s + 3·35-s − 5·37-s − 9·41-s + 49-s − 3·53-s + 2·61-s − 3·65-s + 8·67-s + 6·73-s + 15·85-s + 13·89-s − 91-s + 12·95-s − 97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377·7-s + 0.277·13-s − 1.21·17-s − 0.917·19-s + 0.834·23-s + 4/5·25-s + 1.29·29-s − 0.718·31-s + 0.507·35-s − 0.821·37-s − 1.40·41-s + 1/7·49-s − 0.412·53-s + 0.256·61-s − 0.372·65-s + 0.977·67-s + 0.702·73-s + 1.62·85-s + 1.37·89-s − 0.104·91-s + 1.23·95-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.79057660879984, −13.96354201244825, −13.52733992216730, −12.92825207243562, −12.55382403993092, −11.96708963875910, −11.56928432151139, −10.90005621633281, −10.72002826410773, −10.03316132802229, −9.278725005235848, −8.776766206705312, −8.357151283595622, −7.936523521967934, −7.104250533639562, −6.772874315882011, −6.371856740719255, −5.458435222146220, −4.790311776035320, −4.367715971397635, −3.685376464135956, −3.286922401278732, −2.490251121338398, −1.743234302647602, −0.6931472250502738, 0, 0.6931472250502738, 1.743234302647602, 2.490251121338398, 3.286922401278732, 3.685376464135956, 4.367715971397635, 4.790311776035320, 5.458435222146220, 6.371856740719255, 6.772874315882011, 7.104250533639562, 7.936523521967934, 8.357151283595622, 8.776766206705312, 9.278725005235848, 10.03316132802229, 10.72002826410773, 10.90005621633281, 11.56928432151139, 11.96708963875910, 12.55382403993092, 12.92825207243562, 13.52733992216730, 13.96354201244825, 14.79057660879984

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