Properties

Label 2-1232-1.1-c3-0-58
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  − 10.1·3-s + 8.69·5-s + 7·7-s + 75.9·9-s + 11·11-s − 76.3·13-s − 88.2·15-s + 39.7·17-s + 27.9·19-s − 71.0·21-s − 87.2·23-s − 49.3·25-s − 496.·27-s − 38.3·29-s + 186.·31-s − 111.·33-s + 60.8·35-s − 218.·37-s + 774.·39-s + 80.1·41-s + 35.1·43-s + 660.·45-s + 282.·47-s + 49·49-s − 403.·51-s + 145.·53-s + 95.6·55-s + ⋯
L(s)  = 1  − 1.95·3-s + 0.778·5-s + 0.377·7-s + 2.81·9-s + 0.301·11-s − 1.62·13-s − 1.51·15-s + 0.566·17-s + 0.337·19-s − 0.738·21-s − 0.790·23-s − 0.394·25-s − 3.53·27-s − 0.245·29-s + 1.07·31-s − 0.588·33-s + 0.294·35-s − 0.972·37-s + 3.18·39-s + 0.305·41-s + 0.124·43-s + 2.18·45-s + 0.877·47-s + 0.142·49-s − 1.10·51-s + 0.376·53-s + 0.234·55-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 \( 1 + 10.1T + 27T^{2} \)
5 \( 1 - 8.69T + 125T^{2} \)
13 \( 1 + 76.3T + 2.19e3T^{2} \)
17 \( 1 - 39.7T + 4.91e3T^{2} \)
19 \( 1 - 27.9T + 6.85e3T^{2} \)
23 \( 1 + 87.2T + 1.21e4T^{2} \)
29 \( 1 + 38.3T + 2.43e4T^{2} \)
31 \( 1 - 186.T + 2.97e4T^{2} \)
37 \( 1 + 218.T + 5.06e4T^{2} \)
41 \( 1 - 80.1T + 6.89e4T^{2} \)
43 \( 1 - 35.1T + 7.95e4T^{2} \)
47 \( 1 - 282.T + 1.03e5T^{2} \)
53 \( 1 - 145.T + 1.48e5T^{2} \)
59 \( 1 + 91.0T + 2.05e5T^{2} \)
61 \( 1 - 808.T + 2.26e5T^{2} \)
67 \( 1 + 794.T + 3.00e5T^{2} \)
71 \( 1 + 946.T + 3.57e5T^{2} \)
73 \( 1 - 801.T + 3.89e5T^{2} \)
79 \( 1 - 890.T + 4.93e5T^{2} \)
83 \( 1 - 559.T + 5.71e5T^{2} \)
89 \( 1 + 1.52e3T + 7.04e5T^{2} \)
97 \( 1 - 664.T + 9.12e5T^{2} \)
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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

−9.331365291576990630597634661955, −7.78520653904640543621688212872, −7.08985415769147355126255954123, −6.24664216259771538233751562328, −5.51448642916635854182502317376, −4.97986559769831821620795053815, −4.06157890309676685964490331456, −2.22255664627633354396678006536, −1.16552832259182026736239862710, 0, 1.16552832259182026736239862710, 2.22255664627633354396678006536, 4.06157890309676685964490331456, 4.97986559769831821620795053815, 5.51448642916635854182502317376, 6.24664216259771538233751562328, 7.08985415769147355126255954123, 7.78520653904640543621688212872, 9.331365291576990630597634661955

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