Properties

Label 2-60984-1.1-c1-0-35
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 + 6·17-s + 8·23-s − 25-s − 2·29-s + 4·31-s − 2·35-s − 2·37-s − 2·41-s + 12·43-s − 12·47-s + 49-s − 6·53-s − 4·59-s − 6·61-s + 12·65-s + 4·67-s + 2·73-s − 8·79-s − 12·85-s + 6·89-s − 6·91-s + 18·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 1.66·13-s + 1.45·17-s + 1.66·23-s − 1/5·25-s − 0.371·29-s + 0.718·31-s − 0.338·35-s − 0.328·37-s − 0.312·41-s + 1.82·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s − 0.520·59-s − 0.768·61-s + 1.48·65-s + 0.488·67-s + 0.234·73-s − 0.900·79-s − 1.30·85-s + 0.635·89-s − 0.628·91-s + 1.82·97-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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.66089680362076, −14.20593038064051, −13.60261252547449, −12.74807605240400, −12.59454281248513, −11.92823461616247, −11.66027881863791, −11.04288725957662, −10.53170753323078, −9.882035965224360, −9.491311203893550, −8.913519569921754, −8.160493546456773, −7.724790158097781, −7.424735766042218, −6.878159502738314, −6.119506879988640, −5.377866471033004, −4.895671591422188, −4.509311303076956, −3.676114666824564, −3.109845722830487, −2.577077211785605, −1.646832372549412, −0.8652250927769153, 0, 0.8652250927769153, 1.646832372549412, 2.577077211785605, 3.109845722830487, 3.676114666824564, 4.509311303076956, 4.895671591422188, 5.377866471033004, 6.119506879988640, 6.878159502738314, 7.424735766042218, 7.724790158097781, 8.160493546456773, 8.913519569921754, 9.491311203893550, 9.882035965224360, 10.53170753323078, 11.04288725957662, 11.66027881863791, 11.92823461616247, 12.59454281248513, 12.74807605240400, 13.60261252547449, 14.20593038064051, 14.66089680362076

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