Properties

Label 2-1323-1.1-c1-0-12
Degree $2$
Conductor $1323$
Sign $1$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 3·5-s − 6·10-s − 2·11-s − 6·13-s − 4·16-s + 3·17-s + 6·19-s + 6·20-s + 4·22-s + 8·23-s + 4·25-s + 12·26-s + 2·29-s − 6·31-s + 8·32-s − 6·34-s + 9·37-s − 12·38-s + 9·41-s − 9·43-s − 4·44-s − 16·46-s − 3·47-s − 8·50-s − 12·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.34·5-s − 1.89·10-s − 0.603·11-s − 1.66·13-s − 16-s + 0.727·17-s + 1.37·19-s + 1.34·20-s + 0.852·22-s + 1.66·23-s + 4/5·25-s + 2.35·26-s + 0.371·29-s − 1.07·31-s + 1.41·32-s − 1.02·34-s + 1.47·37-s − 1.94·38-s + 1.40·41-s − 1.37·43-s − 0.603·44-s − 2.35·46-s − 0.437·47-s − 1.13·50-s − 1.66·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9738724035\)
\(L(\frac12)\) \(\approx\) \(0.9738724035\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.479394193370238459103428530459, −9.289581222021464707786603330344, −7.977598033740232460345294759921, −7.44806189292664502395185090799, −6.63733359142638917146855284352, −5.40999706001583504561647394868, −4.90845254944016731114307090228, −2.96534644851181413064575814534, −2.10923233390474501409147906935, −0.913197546741854214881948839395, 0.913197546741854214881948839395, 2.10923233390474501409147906935, 2.96534644851181413064575814534, 4.90845254944016731114307090228, 5.40999706001583504561647394868, 6.63733359142638917146855284352, 7.44806189292664502395185090799, 7.977598033740232460345294759921, 9.289581222021464707786603330344, 9.479394193370238459103428530459

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