Properties

Label 2-1232-1.1-c3-0-2
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  − 5.17·3-s − 10.0·5-s + 7·7-s − 0.259·9-s − 11·11-s − 84.5·13-s + 52.1·15-s − 38.2·17-s + 127.·19-s − 36.1·21-s − 140.·23-s − 23.3·25-s + 140.·27-s − 116.·29-s − 338.·31-s + 56.8·33-s − 70.5·35-s − 75.3·37-s + 437.·39-s − 22.4·41-s − 181.·43-s + 2.61·45-s − 300.·47-s + 49·49-s + 197.·51-s − 31.8·53-s + 110.·55-s + ⋯
L(s)  = 1  − 0.995·3-s − 0.901·5-s + 0.377·7-s − 0.00960·9-s − 0.301·11-s − 1.80·13-s + 0.897·15-s − 0.545·17-s + 1.53·19-s − 0.376·21-s − 1.27·23-s − 0.186·25-s + 1.00·27-s − 0.747·29-s − 1.96·31-s + 0.300·33-s − 0.340·35-s − 0.334·37-s + 1.79·39-s − 0.0854·41-s − 0.644·43-s + 0.00865·45-s − 0.932·47-s + 0.142·49-s + 0.543·51-s − 0.0825·53-s + 0.271·55-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

\(L(2)\) \(\approx\) \(0.1893226814\)
\(L(\frac12)\) \(\approx\) \(0.1893226814\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.17T + 27T^{2} \)
5 \( 1 + 10.0T + 125T^{2} \)
13 \( 1 + 84.5T + 2.19e3T^{2} \)
17 \( 1 + 38.2T + 4.91e3T^{2} \)
19 \( 1 - 127.T + 6.85e3T^{2} \)
23 \( 1 + 140.T + 1.21e4T^{2} \)
29 \( 1 + 116.T + 2.43e4T^{2} \)
31 \( 1 + 338.T + 2.97e4T^{2} \)
37 \( 1 + 75.3T + 5.06e4T^{2} \)
41 \( 1 + 22.4T + 6.89e4T^{2} \)
43 \( 1 + 181.T + 7.95e4T^{2} \)
47 \( 1 + 300.T + 1.03e5T^{2} \)
53 \( 1 + 31.8T + 1.48e5T^{2} \)
59 \( 1 - 68.3T + 2.05e5T^{2} \)
61 \( 1 + 145.T + 2.26e5T^{2} \)
67 \( 1 - 668.T + 3.00e5T^{2} \)
71 \( 1 + 727.T + 3.57e5T^{2} \)
73 \( 1 + 416.T + 3.89e5T^{2} \)
79 \( 1 + 458.T + 4.93e5T^{2} \)
83 \( 1 + 355.T + 5.71e5T^{2} \)
89 \( 1 + 1.24e3T + 7.04e5T^{2} \)
97 \( 1 + 935.T + 9.12e5T^{2} \)
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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

−9.510395971316266183628219458135, −8.374612613641207276614766146335, −7.50077024031823612106992049921, −7.09601242843573675554713544111, −5.74798232953495470437783446747, −5.18772456039638925425717161194, −4.35433900103875526129779816371, −3.21678272456601636832384188498, −1.90720178597180734805306218556, −0.22214449369852328533142672289, 0.22214449369852328533142672289, 1.90720178597180734805306218556, 3.21678272456601636832384188498, 4.35433900103875526129779816371, 5.18772456039638925425717161194, 5.74798232953495470437783446747, 7.09601242843573675554713544111, 7.50077024031823612106992049921, 8.374612613641207276614766146335, 9.510395971316266183628219458135

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