Properties

Label 2-924-1.1-c1-0-6
Degree $2$
Conductor $924$
Sign $-1$
Analytic cond. $7.37817$
Root an. cond. $2.71628$
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-s − 3·5-s − 7-s + 9-s − 11-s + 3·13-s − 3·15-s − 2·17-s − 3·19-s − 21-s − 4·23-s + 4·25-s + 27-s − 9·29-s − 2·31-s − 33-s + 3·35-s − 11·37-s + 3·39-s − 4·41-s − 4·43-s − 3·45-s − 3·47-s + 49-s − 2·51-s − 4·53-s + 3·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 0.774·15-s − 0.485·17-s − 0.688·19-s − 0.218·21-s − 0.834·23-s + 4/5·25-s + 0.192·27-s − 1.67·29-s − 0.359·31-s − 0.174·33-s + 0.507·35-s − 1.80·37-s + 0.480·39-s − 0.624·41-s − 0.609·43-s − 0.447·45-s − 0.437·47-s + 1/7·49-s − 0.280·51-s − 0.549·53-s + 0.404·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 924 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 924 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(924\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(7.37817\)
Root analytic conductor: \(2.71628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 924,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−9.550209289339973999946918010751, −8.607584234319820279843149378887, −8.112770171371127759402374804737, −7.24264125802517025269438845065, −6.40761147031881701581107354593, −5.11675518536581268064805822554, −3.85808269239208963151847638821, −3.53426831848528683359566672044, −2.00543267362711116052566627685, 0, 2.00543267362711116052566627685, 3.53426831848528683359566672044, 3.85808269239208963151847638821, 5.11675518536581268064805822554, 6.40761147031881701581107354593, 7.24264125802517025269438845065, 8.112770171371127759402374804737, 8.607584234319820279843149378887, 9.550209289339973999946918010751

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