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Properties

Label 2-1323-63.25-c1-0-6
Degree $2$
Conductor $1323$
Sign $-0.403 - 0.914i$
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  + 1.10·2-s − 0.783·4-s + (−0.0527 + 0.0913i)5-s − 3.07·8-s + (−0.0581 + 0.100i)10-s + (1.66 + 2.89i)11-s + (1.23 + 2.14i)13-s − 1.81·16-s + (−0.806 + 1.39i)17-s + (−3.84 − 6.65i)19-s + (0.0413 − 0.0715i)20-s + (1.84 + 3.18i)22-s + (−0.948 + 1.64i)23-s + (2.49 + 4.32i)25-s + (1.36 + 2.36i)26-s + ⋯
L(s)  = 1  + 0.779·2-s − 0.391·4-s + (−0.0235 + 0.0408i)5-s − 1.08·8-s + (−0.0183 + 0.0318i)10-s + (0.503 + 0.871i)11-s + (0.343 + 0.595i)13-s − 0.454·16-s + (−0.195 + 0.338i)17-s + (−0.881 − 1.52i)19-s + (0.00924 − 0.0160i)20-s + (0.392 + 0.679i)22-s + (−0.197 + 0.342i)23-s + (0.498 + 0.864i)25-s + (0.268 + 0.464i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.403 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.403 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.403 - 0.914i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.403 - 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.257505515\)
\(L(\frac12)\) \(\approx\) \(1.257505515\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.10T + 2T^{2} \)
5 \( 1 + (0.0527 - 0.0913i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.66 - 2.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.23 - 2.14i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.806 - 1.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.84 + 6.65i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.948 - 1.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.64 - 8.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.26T + 31T^{2} \)
37 \( 1 + (-0.991 - 1.71i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.74 - 6.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.77 - 6.53i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.19T + 47T^{2} \)
53 \( 1 + (4.98 - 8.64i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.45T + 59T^{2} \)
61 \( 1 - 5.67T + 61T^{2} \)
67 \( 1 - 9.97T + 67T^{2} \)
71 \( 1 + 3.29T + 71T^{2} \)
73 \( 1 + (2.36 - 4.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 7.69T + 79T^{2} \)
83 \( 1 + (0.584 - 1.01i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.01 + 5.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.90 + 3.29i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.569913544126376016460475982668, −9.215144778428729653154092497340, −8.436249506984087006100046128490, −7.15126291674649334755856448158, −6.59461763992660666003311319470, −5.52924973709190328140810347416, −4.69888227704184976198082562913, −4.02687349161366695991013170325, −3.04597938543634563067500696252, −1.66096603512996230208087014845, 0.40075529898760709692072202088, 2.22648014103545884837887586020, 3.66115274321393244203064182287, 3.93878284182369899613133546362, 5.23369505075258632654474880886, 5.88605498695436600610478103842, 6.57502551847360353638763645114, 7.931597512110091923302277667583, 8.535048137364408564264523297992, 9.296562742583092269419068876662

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