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Properties

Label 2-1323-63.47-c1-0-30
Degree $2$
Conductor $1323$
Sign $-0.0832 + 0.996i$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.34 + 1.35i)2-s + (2.66 − 4.62i)4-s + 1.20·5-s + 9.04i·8-s + (−2.82 + 1.62i)10-s − 2.48i·11-s + (1.63 − 0.942i)13-s + (−6.90 − 11.9i)16-s + (−0.601 − 1.04i)17-s + (−6.46 − 3.73i)19-s + (3.21 − 5.56i)20-s + (3.36 + 5.83i)22-s − 3.04i·23-s − 3.55·25-s + (−2.55 + 4.42i)26-s + ⋯
L(s)  = 1  + (−1.65 + 0.957i)2-s + (1.33 − 2.31i)4-s + 0.537·5-s + 3.19i·8-s + (−0.892 + 0.515i)10-s − 0.749i·11-s + (0.452 − 0.261i)13-s + (−1.72 − 2.99i)16-s + (−0.145 − 0.252i)17-s + (−1.48 − 0.856i)19-s + (0.717 − 1.24i)20-s + (0.718 + 1.24i)22-s − 0.634i·23-s − 0.710·25-s + (−0.500 + 0.867i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0832 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0832 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.0832 + 0.996i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (656, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.0832 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3453244029\)
\(L(\frac12)\) \(\approx\) \(0.3453244029\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.34 - 1.35i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 1.20T + 5T^{2} \)
11 \( 1 + 2.48iT - 11T^{2} \)
13 \( 1 + (-1.63 + 0.942i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.601 + 1.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.46 + 3.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.04iT - 23T^{2} \)
29 \( 1 + (-0.173 - 0.100i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.03 - 1.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.865 - 1.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.36 + 5.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.00656 + 0.0113i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.717 - 1.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.58 - 4.95i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.10 - 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.73 - 5.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.57 + 4.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.0iT - 71T^{2} \)
73 \( 1 + (7.51 - 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.74 + 4.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.60 - 2.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.98 - 6.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.06 - 1.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.103334695271131446845161633139, −8.702996842115368584159040646259, −7.982007076064721472656320784530, −7.03966298042798544748687207705, −6.25724883297710631227327568936, −5.80608699659539759550424781851, −4.61495999112352646683248007088, −2.74607970917816640694644349333, −1.59541359222922364491488385160, −0.24116064665156245399068416654, 1.56471900366141507989201108767, 2.13990190366062183843677542544, 3.39888268803495135798055173843, 4.38931835033245413572963162043, 6.07324597030196889693259736330, 6.79821240392743005842018867808, 7.83921660305713200805116159555, 8.362734051487394546050718214687, 9.249461981562054810829243644854, 9.848891909893762561620249051606

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