Properties

Label 2-192-1.1-c7-0-15
Degree $2$
Conductor $192$
Sign $-1$
Analytic cond. $59.9779$
Root an. cond. $7.74454$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27·3-s − 390·5-s + 64·7-s + 729·9-s − 948·11-s + 5.09e3·13-s + 1.05e4·15-s + 2.83e4·17-s − 8.62e3·19-s − 1.72e3·21-s + 1.52e4·23-s + 7.39e4·25-s − 1.96e4·27-s − 3.65e4·29-s + 2.76e5·31-s + 2.55e4·33-s − 2.49e4·35-s − 2.68e5·37-s − 1.37e5·39-s − 6.29e5·41-s + 6.85e5·43-s − 2.84e5·45-s − 5.83e5·47-s − 8.19e5·49-s − 7.66e5·51-s + 4.28e5·53-s + 3.69e5·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.39·5-s + 0.0705·7-s + 1/3·9-s − 0.214·11-s + 0.643·13-s + 0.805·15-s + 1.40·17-s − 0.288·19-s − 0.0407·21-s + 0.262·23-s + 0.946·25-s − 0.192·27-s − 0.277·29-s + 1.66·31-s + 0.123·33-s − 0.0984·35-s − 0.871·37-s − 0.371·39-s − 1.42·41-s + 1.31·43-s − 0.465·45-s − 0.819·47-s − 0.995·49-s − 0.809·51-s + 0.394·53-s + 0.299·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-1$
Analytic conductor: \(59.9779\)
Root analytic conductor: \(7.74454\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 192,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 + p^{3} T \)
good5 \( 1 + 78 p T + p^{7} T^{2} \)
7 \( 1 - 64 T + p^{7} T^{2} \)
11 \( 1 + 948 T + p^{7} T^{2} \)
13 \( 1 - 5098 T + p^{7} T^{2} \)
17 \( 1 - 28386 T + p^{7} T^{2} \)
19 \( 1 + 8620 T + p^{7} T^{2} \)
23 \( 1 - 15288 T + p^{7} T^{2} \)
29 \( 1 + 36510 T + p^{7} T^{2} \)
31 \( 1 - 276808 T + p^{7} T^{2} \)
37 \( 1 + 268526 T + p^{7} T^{2} \)
41 \( 1 + 629718 T + p^{7} T^{2} \)
43 \( 1 - 685772 T + p^{7} T^{2} \)
47 \( 1 + 583296 T + p^{7} T^{2} \)
53 \( 1 - 428058 T + p^{7} T^{2} \)
59 \( 1 - 1306380 T + p^{7} T^{2} \)
61 \( 1 + 300662 T + p^{7} T^{2} \)
67 \( 1 + 507244 T + p^{7} T^{2} \)
71 \( 1 + 5560632 T + p^{7} T^{2} \)
73 \( 1 - 1369082 T + p^{7} T^{2} \)
79 \( 1 - 6913720 T + p^{7} T^{2} \)
83 \( 1 + 4376748 T + p^{7} T^{2} \)
89 \( 1 + 8528310 T + p^{7} T^{2} \)
97 \( 1 + 8826814 T + p^{7} T^{2} \)
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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

−10.92855235211862053863189827700, −9.974495480191317072657874708298, −8.481232644464964717831537732914, −7.74663214224141048740806794350, −6.66490151997038779333662745852, −5.37917673145249050354010592240, −4.22120609695061456796833116405, −3.19349468104231142538713506612, −1.18240424046115842984692125562, 0, 1.18240424046115842984692125562, 3.19349468104231142538713506612, 4.22120609695061456796833116405, 5.37917673145249050354010592240, 6.66490151997038779333662745852, 7.74663214224141048740806794350, 8.481232644464964717831537732914, 9.974495480191317072657874708298, 10.92855235211862053863189827700

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