Properties

Label 2-1232-1.1-c3-0-51
Degree $2$
Conductor $1232$
Sign $1$
Analytic cond. $72.6903$
Root an. cond. $8.52586$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.70·3-s + 5.23·5-s + 7·7-s + 32.4·9-s + 11·11-s − 29.3·13-s + 40.3·15-s + 107.·17-s − 34.2·19-s + 53.9·21-s − 10.3·23-s − 97.5·25-s + 41.7·27-s + 268.·29-s + 300.·31-s + 84.7·33-s + 36.6·35-s − 213.·37-s − 226.·39-s + 481.·41-s − 447.·43-s + 169.·45-s + 335.·47-s + 49·49-s + 830.·51-s + 183.·53-s + 57.5·55-s + ⋯
L(s)  = 1  + 1.48·3-s + 0.468·5-s + 0.377·7-s + 1.20·9-s + 0.301·11-s − 0.626·13-s + 0.694·15-s + 1.53·17-s − 0.414·19-s + 0.560·21-s − 0.0941·23-s − 0.780·25-s + 0.297·27-s + 1.72·29-s + 1.73·31-s + 0.447·33-s + 0.177·35-s − 0.947·37-s − 0.928·39-s + 1.83·41-s − 1.58·43-s + 0.562·45-s + 1.04·47-s + 0.142·49-s + 2.28·51-s + 0.476·53-s + 0.141·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(72.6903\)
Root analytic conductor: \(8.52586\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.653724345\)
\(L(\frac12)\) \(\approx\) \(4.653724345\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - 7T \)
11 \( 1 - 11T \)
good3 \( 1 - 7.70T + 27T^{2} \)
5 \( 1 - 5.23T + 125T^{2} \)
13 \( 1 + 29.3T + 2.19e3T^{2} \)
17 \( 1 - 107.T + 4.91e3T^{2} \)
19 \( 1 + 34.2T + 6.85e3T^{2} \)
23 \( 1 + 10.3T + 1.21e4T^{2} \)
29 \( 1 - 268.T + 2.43e4T^{2} \)
31 \( 1 - 300.T + 2.97e4T^{2} \)
37 \( 1 + 213.T + 5.06e4T^{2} \)
41 \( 1 - 481.T + 6.89e4T^{2} \)
43 \( 1 + 447.T + 7.95e4T^{2} \)
47 \( 1 - 335.T + 1.03e5T^{2} \)
53 \( 1 - 183.T + 1.48e5T^{2} \)
59 \( 1 - 58.1T + 2.05e5T^{2} \)
61 \( 1 + 624.T + 2.26e5T^{2} \)
67 \( 1 - 226.T + 3.00e5T^{2} \)
71 \( 1 - 1.12e3T + 3.57e5T^{2} \)
73 \( 1 + 87.3T + 3.89e5T^{2} \)
79 \( 1 - 485.T + 4.93e5T^{2} \)
83 \( 1 - 884.T + 5.71e5T^{2} \)
89 \( 1 + 1.28e3T + 7.04e5T^{2} \)
97 \( 1 + 49.2T + 9.12e5T^{2} \)
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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

−9.401989796773484493786141131589, −8.337832938183471724111819240333, −8.034646372460423703545050656148, −7.07696976599093434256655544665, −6.05416774632017833803569543200, −4.95415330407543640987994503600, −3.96578346283810094851852650541, −2.96874686061053410119953447341, −2.21449998889540270345320816908, −1.09856246365125948909741431018, 1.09856246365125948909741431018, 2.21449998889540270345320816908, 2.96874686061053410119953447341, 3.96578346283810094851852650541, 4.95415330407543640987994503600, 6.05416774632017833803569543200, 7.07696976599093434256655544665, 8.034646372460423703545050656148, 8.337832938183471724111819240333, 9.401989796773484493786141131589

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