Properties

Label 2-1232-1.1-c3-0-72
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.49·3-s − 16.0·5-s + 7·7-s + 3.16·9-s + 11·11-s + 35.3·13-s − 88.4·15-s + 40.4·17-s − 118.·19-s + 38.4·21-s + 174.·23-s + 134.·25-s − 130.·27-s − 262.·29-s + 36.1·31-s + 60.4·33-s − 112.·35-s + 19.0·37-s + 194.·39-s + 156.·41-s − 287.·43-s − 50.9·45-s − 397.·47-s + 49·49-s + 222.·51-s + 272.·53-s − 177.·55-s + ⋯
L(s)  = 1  + 1.05·3-s − 1.43·5-s + 0.377·7-s + 0.117·9-s + 0.301·11-s + 0.754·13-s − 1.52·15-s + 0.577·17-s − 1.42·19-s + 0.399·21-s + 1.58·23-s + 1.07·25-s − 0.933·27-s − 1.68·29-s + 0.209·31-s + 0.318·33-s − 0.544·35-s + 0.0846·37-s + 0.797·39-s + 0.598·41-s − 1.01·43-s − 0.168·45-s − 1.23·47-s + 0.142·49-s + 0.610·51-s + 0.706·53-s − 0.434·55-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: $\chi_{1232} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1232,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.49T + 27T^{2} \)
5 \( 1 + 16.0T + 125T^{2} \)
13 \( 1 - 35.3T + 2.19e3T^{2} \)
17 \( 1 - 40.4T + 4.91e3T^{2} \)
19 \( 1 + 118.T + 6.85e3T^{2} \)
23 \( 1 - 174.T + 1.21e4T^{2} \)
29 \( 1 + 262.T + 2.43e4T^{2} \)
31 \( 1 - 36.1T + 2.97e4T^{2} \)
37 \( 1 - 19.0T + 5.06e4T^{2} \)
41 \( 1 - 156.T + 6.89e4T^{2} \)
43 \( 1 + 287.T + 7.95e4T^{2} \)
47 \( 1 + 397.T + 1.03e5T^{2} \)
53 \( 1 - 272.T + 1.48e5T^{2} \)
59 \( 1 - 507.T + 2.05e5T^{2} \)
61 \( 1 - 35.5T + 2.26e5T^{2} \)
67 \( 1 + 979.T + 3.00e5T^{2} \)
71 \( 1 + 750.T + 3.57e5T^{2} \)
73 \( 1 - 395.T + 3.89e5T^{2} \)
79 \( 1 - 736.T + 4.93e5T^{2} \)
83 \( 1 + 582.T + 5.71e5T^{2} \)
89 \( 1 + 806.T + 7.04e5T^{2} \)
97 \( 1 + 957.T + 9.12e5T^{2} \)
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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

−8.683996258166581349922253356726, −8.237863250077752008599130632360, −7.54566676174316696970623319162, −6.71143609886022583311250348305, −5.44761148624790900441710460375, −4.23548561772427339989478905749, −3.67309679932694440419065489878, −2.80655585984836223741936477997, −1.46557721086066685720925114035, 0, 1.46557721086066685720925114035, 2.80655585984836223741936477997, 3.67309679932694440419065489878, 4.23548561772427339989478905749, 5.44761148624790900441710460375, 6.71143609886022583311250348305, 7.54566676174316696970623319162, 8.237863250077752008599130632360, 8.683996258166581349922253356726

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