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Properties

Label 2-1323-63.58-c1-0-12
Degree $2$
Conductor $1323$
Sign $-0.211 - 0.977i$
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  + 1.72·2-s + 0.981·4-s + (1.75 + 3.04i)5-s − 1.75·8-s + (3.03 + 5.25i)10-s + (−3.04 + 5.27i)11-s + (0.560 − 0.970i)13-s − 4.99·16-s + (0.601 + 1.04i)17-s + (1.10 − 1.90i)19-s + (1.72 + 2.98i)20-s + (−5.25 + 9.10i)22-s + (−0.636 − 1.10i)23-s + (−3.66 + 6.35i)25-s + (0.967 − 1.67i)26-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.490·4-s + (0.785 + 1.36i)5-s − 0.621·8-s + (0.958 + 1.66i)10-s + (−0.918 + 1.59i)11-s + (0.155 − 0.269i)13-s − 1.24·16-s + (0.146 + 0.252i)17-s + (0.252 − 0.438i)19-s + (0.385 + 0.667i)20-s + (−1.12 + 1.94i)22-s + (−0.132 − 0.229i)23-s + (−0.733 + 1.27i)25-s + (0.189 − 0.328i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.211 - 0.977i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.760059464\)
\(L(\frac12)\) \(\approx\) \(2.760059464\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.72T + 2T^{2} \)
5 \( 1 + (-1.75 - 3.04i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.04 - 5.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.560 + 0.970i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.601 - 1.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.10 + 1.90i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.636 + 1.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.10 - 5.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.188T + 31T^{2} \)
37 \( 1 + (1.78 - 3.09i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.68 - 2.91i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.90 + 3.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.72T + 47T^{2} \)
53 \( 1 + (4.16 + 7.22i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + 7.91T + 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 + (-2.65 - 4.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 9.21T + 79T^{2} \)
83 \( 1 + (0.624 + 1.08i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.77 + 4.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.24 - 14.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.11682511067669216311419905901, −9.274015327336481536690666265832, −8.044893719797079152495292447936, −6.90638879165312850347417411317, −6.63374531221838924790851617842, −5.45942807722319640102096539950, −4.95868314283439848415223688426, −3.77403716299840291834089634293, −2.82365890045954564887674232656, −2.14173289648366085315521923617, 0.73523228776939836875863801820, 2.31720502841294102843132128429, 3.42749546467542114960308275462, 4.38740852577616428608241999930, 5.37947142268476639408608574708, 5.60319345038066823772957506955, 6.45568768045914006539750100312, 7.933219163680651927903807493775, 8.642530282833772824521134549017, 9.283594336321026905626010451107

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