Properties

Label 2-1449-1.1-c1-0-32
Degree $2$
Conductor $1449$
Sign $1$
Analytic cond. $11.5703$
Root an. cond. $3.40151$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30·2-s + 3.30·4-s + 0.697·5-s − 7-s + 3.00·8-s + 1.60·10-s + 5·11-s + 2.30·13-s − 2.30·14-s + 0.302·16-s + 5.60·17-s − 1.60·19-s + 2.30·20-s + 11.5·22-s − 23-s − 4.51·25-s + 5.30·26-s − 3.30·28-s − 6.21·29-s + 3·31-s − 5.30·32-s + 12.9·34-s − 0.697·35-s − 9·37-s − 3.69·38-s + 2.09·40-s + 12.2·41-s + ⋯
L(s)  = 1  + 1.62·2-s + 1.65·4-s + 0.311·5-s − 0.377·7-s + 1.06·8-s + 0.507·10-s + 1.50·11-s + 0.638·13-s − 0.615·14-s + 0.0756·16-s + 1.35·17-s − 0.368·19-s + 0.514·20-s + 2.45·22-s − 0.208·23-s − 0.902·25-s + 1.03·26-s − 0.624·28-s − 1.15·29-s + 0.538·31-s − 0.937·32-s + 2.21·34-s − 0.117·35-s − 1.47·37-s − 0.599·38-s + 0.330·40-s + 1.90·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(11.5703\)
Root analytic conductor: \(3.40151\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1449,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.720814084\)
\(L(\frac12)\) \(\approx\) \(4.720814084\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
23 \( 1 + T \)
good2 \( 1 - 2.30T + 2T^{2} \)
5 \( 1 - 0.697T + 5T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 2.30T + 13T^{2} \)
17 \( 1 - 5.60T + 17T^{2} \)
19 \( 1 + 1.60T + 19T^{2} \)
29 \( 1 + 6.21T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + 9T + 37T^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 - 5.51T + 43T^{2} \)
47 \( 1 - 8.60T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + 3.90T + 59T^{2} \)
61 \( 1 + 1.09T + 61T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 + 0.908T + 71T^{2} \)
73 \( 1 + 2.21T + 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + 5.60T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 + 17.6T + 97T^{2} \)
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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

−9.514036635781588564714737889540, −8.852027958542071860506485703101, −7.55418748929855667783714984233, −6.76043834365634876362716626914, −5.84653261443195908566468031331, −5.63805356577728493880087378357, −4.09784331485556271733378342384, −3.86851547490128368290759093466, −2.73947671800829403837261009406, −1.46792622502757620795937226518, 1.46792622502757620795937226518, 2.73947671800829403837261009406, 3.86851547490128368290759093466, 4.09784331485556271733378342384, 5.63805356577728493880087378357, 5.84653261443195908566468031331, 6.76043834365634876362716626914, 7.55418748929855667783714984233, 8.852027958542071860506485703101, 9.514036635781588564714737889540

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