Properties

Label 2-1323-1.1-c1-0-4
Degree $2$
Conductor $1323$
Sign $1$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
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  + 0.760·2-s − 1.42·4-s − 3.18·5-s − 2.60·8-s − 2.42·10-s + 2.23·11-s − 3.70·13-s + 0.861·16-s + 5.60·17-s − 4.42·19-s + 4.52·20-s + 1.70·22-s − 0.942·23-s + 5.12·25-s − 2.81·26-s + 10.1·29-s + 5.70·31-s + 5.86·32-s + 4.26·34-s + 3.12·37-s − 3.36·38-s + 8.28·40-s − 3.98·41-s − 3.28·43-s − 3.18·44-s − 0.717·46-s + 0.225·47-s + ⋯
L(s)  = 1  + 0.538·2-s − 0.710·4-s − 1.42·5-s − 0.920·8-s − 0.765·10-s + 0.675·11-s − 1.02·13-s + 0.215·16-s + 1.35·17-s − 1.01·19-s + 1.01·20-s + 0.363·22-s − 0.196·23-s + 1.02·25-s − 0.552·26-s + 1.88·29-s + 1.02·31-s + 1.03·32-s + 0.731·34-s + 0.513·37-s − 0.545·38-s + 1.30·40-s − 0.622·41-s − 0.500·43-s − 0.479·44-s − 0.105·46-s + 0.0328·47-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.132077402\)
\(L(\frac12)\) \(\approx\) \(1.132077402\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.760T + 2T^{2} \)
5 \( 1 + 3.18T + 5T^{2} \)
11 \( 1 - 2.23T + 11T^{2} \)
13 \( 1 + 3.70T + 13T^{2} \)
17 \( 1 - 5.60T + 17T^{2} \)
19 \( 1 + 4.42T + 19T^{2} \)
23 \( 1 + 0.942T + 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 - 5.70T + 31T^{2} \)
37 \( 1 - 3.12T + 37T^{2} \)
41 \( 1 + 3.98T + 41T^{2} \)
43 \( 1 + 3.28T + 43T^{2} \)
47 \( 1 - 0.225T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 - 2.05T + 59T^{2} \)
61 \( 1 - 5.84T + 61T^{2} \)
67 \( 1 - 7.42T + 67T^{2} \)
71 \( 1 - 7.26T + 71T^{2} \)
73 \( 1 + 7.55T + 73T^{2} \)
79 \( 1 + 6.82T + 79T^{2} \)
83 \( 1 - 8.11T + 83T^{2} \)
89 \( 1 + 9.72T + 89T^{2} \)
97 \( 1 + 0.842T + 97T^{2} \)
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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

−9.727285742764224034032361197842, −8.510972566094292981428204874845, −8.221145893715625897739793843526, −7.19618995858780359176017427665, −6.30451642111038210770496949945, −5.14364003819429536585102039665, −4.40694138984125799620544291856, −3.76575394111948690607787978118, −2.80627788180869429299571593262, −0.71654060367547032107368546326, 0.71654060367547032107368546326, 2.80627788180869429299571593262, 3.76575394111948690607787978118, 4.40694138984125799620544291856, 5.14364003819429536585102039665, 6.30451642111038210770496949945, 7.19618995858780359176017427665, 8.221145893715625897739793843526, 8.510972566094292981428204874845, 9.727285742764224034032361197842

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