Properties

Label 2-1232-308.307-c2-0-69
Degree $2$
Conductor $1232$
Sign $1$
Analytic cond. $33.5695$
Root an. cond. $5.79392$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.96·3-s − 7·7-s + 26.5·9-s − 11·11-s + 8.64·13-s + 27.1·17-s − 41.7·21-s + 25·25-s + 104.·27-s + 35.1·31-s − 65.5·33-s − 52.6·37-s + 51.5·39-s + 72.1·41-s − 52.6·43-s + 19.0·47-s + 49·49-s + 161.·51-s − 52.6·53-s − 28.3·59-s + 106.·61-s − 185.·63-s − 99.8·73-s + 149.·75-s + 77·77-s + 157.·79-s + 384.·81-s + ⋯
L(s)  = 1  + 1.98·3-s − 7-s + 2.94·9-s − 11-s + 0.665·13-s + 1.59·17-s − 1.98·21-s + 25-s + 3.87·27-s + 1.13·31-s − 1.98·33-s − 1.42·37-s + 1.32·39-s + 1.75·41-s − 1.22·43-s + 0.405·47-s + 0.999·49-s + 3.17·51-s − 0.993·53-s − 0.479·59-s + 1.74·61-s − 2.94·63-s − 1.36·73-s + 1.98·75-s + 77-s + 1.99·79-s + 4.75·81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(33.5695\)
Root analytic conductor: \(5.79392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (1231, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.031085153\)
\(L(\frac12)\) \(\approx\) \(4.031085153\)
\(L(2)\) 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 \( 1 - 5.96T + 9T^{2} \)
5 \( 1 - 25T^{2} \)
13 \( 1 - 8.64T + 169T^{2} \)
17 \( 1 - 27.1T + 289T^{2} \)
19 \( 1 - 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 35.1T + 961T^{2} \)
37 \( 1 + 52.6T + 1.36e3T^{2} \)
41 \( 1 - 72.1T + 1.68e3T^{2} \)
43 \( 1 + 52.6T + 1.84e3T^{2} \)
47 \( 1 - 19.0T + 2.20e3T^{2} \)
53 \( 1 + 52.6T + 2.80e3T^{2} \)
59 \( 1 + 28.3T + 3.48e3T^{2} \)
61 \( 1 - 106.T + 3.72e3T^{2} \)
67 \( 1 - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 99.8T + 5.32e3T^{2} \)
79 \( 1 - 157.T + 6.24e3T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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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

−9.474647274123102285653628143344, −8.643109966119129767690224470202, −8.038007389043707786323139492700, −7.32645246329711362584354092032, −6.45633757927332486480338454369, −5.14120241212597921199381859200, −3.89955662496109751248162859781, −3.17385597132120420944542159349, −2.58368150229558019645112819919, −1.18068152680698404969880027701, 1.18068152680698404969880027701, 2.58368150229558019645112819919, 3.17385597132120420944542159349, 3.89955662496109751248162859781, 5.14120241212597921199381859200, 6.45633757927332486480338454369, 7.32645246329711362584354092032, 8.038007389043707786323139492700, 8.643109966119129767690224470202, 9.474647274123102285653628143344

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