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Properties

Label 2-1323-63.4-c1-0-15
Degree $2$
Conductor $1323$
Sign $0.591 - 0.806i$
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  + (−0.920 + 1.59i)2-s + (−0.695 − 1.20i)4-s + 1.33·5-s − 1.12·8-s + (−1.22 + 2.12i)10-s − 1.51·11-s + (2.58 − 4.48i)13-s + (2.42 − 4.19i)16-s + (0.774 − 1.34i)17-s + (1.25 + 2.16i)19-s + (−0.927 − 1.60i)20-s + (1.39 − 2.41i)22-s + 7.36·23-s − 3.21·25-s + (4.76 + 8.25i)26-s + ⋯
L(s)  = 1  + (−0.650 + 1.12i)2-s + (−0.347 − 0.601i)4-s + 0.596·5-s − 0.396·8-s + (−0.388 + 0.673i)10-s − 0.456·11-s + (0.717 − 1.24i)13-s + (0.605 − 1.04i)16-s + (0.187 − 0.325i)17-s + (0.287 + 0.497i)19-s + (−0.207 − 0.359i)20-s + (0.296 − 0.514i)22-s + 1.53·23-s − 0.643·25-s + (0.934 + 1.61i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.591 - 0.806i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.591 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.260763612\)
\(L(\frac12)\) \(\approx\) \(1.260763612\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.920 - 1.59i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 1.33T + 5T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
13 \( 1 + (-2.58 + 4.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.774 + 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.25 - 2.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.36T + 23T^{2} \)
29 \( 1 + (-0.0309 - 0.0536i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.92 + 3.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.281 + 0.487i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.51 + 7.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.09 - 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.75 + 8.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.755 - 1.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.22 - 7.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.61 + 2.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 + 6.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (-1.37 + 2.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.95 + 5.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.80 - 4.85i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.703 - 1.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.09 - 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.462071688487330443932460999253, −8.895702634355318076773403902432, −7.933077267939894324472299648524, −7.51398098852205824236594777606, −6.50104150305816385753358293253, −5.68966032211403238290773138701, −5.24418460778098954461048636412, −3.60791196902328252221210596101, −2.56409677723269702711564892018, −0.809517273114771430126286713195, 1.07912701134769507157136257747, 2.06739179871099086311785061630, 3.00983386344282705677436880061, 4.07995719212366334072718440314, 5.31142566830706675939626641765, 6.20944323558721258586676560565, 7.07831406050832014092649844975, 8.251562521731169564399313011401, 9.118010214248504939085976857428, 9.437002160077886943287222531192

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