Properties

Label 2-60984-1.1-c1-0-7
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s + 6·13-s + 3·17-s + 4·19-s − 6·23-s − 4·25-s − 6·31-s + 35-s − 6·37-s + 10·41-s − 11·43-s + 9·47-s + 49-s + 12·53-s − 7·59-s − 2·61-s − 6·65-s − 9·67-s + 2·71-s + 2·73-s − 4·79-s + 3·83-s − 3·85-s − 15·89-s − 6·91-s − 4·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 1.66·13-s + 0.727·17-s + 0.917·19-s − 1.25·23-s − 4/5·25-s − 1.07·31-s + 0.169·35-s − 0.986·37-s + 1.56·41-s − 1.67·43-s + 1.31·47-s + 1/7·49-s + 1.64·53-s − 0.911·59-s − 0.256·61-s − 0.744·65-s − 1.09·67-s + 0.237·71-s + 0.234·73-s − 0.450·79-s + 0.329·83-s − 0.325·85-s − 1.58·89-s − 0.628·91-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: \(0\)
Selberg data: \((2,\ 60984,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.862984509\)
\(L(\frac12)\) \(\approx\) \(1.862984509\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−14.12101341483508, −13.73041525048049, −13.49133100723893, −12.69890644077895, −12.23102070170504, −11.80841168283696, −11.27711294793416, −10.77262196624219, −10.20694889357272, −9.728691229601231, −9.128454378895268, −8.558590092055898, −8.152553021372924, −7.402857767488705, −7.204811924212936, −6.246858855257943, −5.811302925629612, −5.507151352928439, −4.534954651443044, −3.804923151841062, −3.623105215636101, −2.937325476552802, −1.968318550580053, −1.329465345252814, −0.4873494132897363, 0.4873494132897363, 1.329465345252814, 1.968318550580053, 2.937325476552802, 3.623105215636101, 3.804923151841062, 4.534954651443044, 5.507151352928439, 5.811302925629612, 6.246858855257943, 7.204811924212936, 7.402857767488705, 8.152553021372924, 8.558590092055898, 9.128454378895268, 9.728691229601231, 10.20694889357272, 10.77262196624219, 11.27711294793416, 11.80841168283696, 12.23102070170504, 12.69890644077895, 13.49133100723893, 13.73041525048049, 14.12101341483508

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