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Properties

Label 2-1323-63.38-c1-0-32
Degree $2$
Conductor $1323$
Sign $-0.282 + 0.959i$
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  − 0.981i·2-s + 1.03·4-s + (0.940 − 1.62i)5-s − 2.98i·8-s + (−1.59 − 0.923i)10-s + (3.54 − 2.04i)11-s + (3.51 − 2.02i)13-s − 0.852·16-s + (0.810 − 1.40i)17-s + (−7.03 + 4.06i)19-s + (0.974 − 1.68i)20-s + (−2.00 − 3.47i)22-s + (3.73 + 2.15i)23-s + (0.730 + 1.26i)25-s + (−1.99 − 3.44i)26-s + ⋯
L(s)  = 1  − 0.694i·2-s + 0.518·4-s + (0.420 − 0.728i)5-s − 1.05i·8-s + (−0.505 − 0.291i)10-s + (1.06 − 0.616i)11-s + (0.974 − 0.562i)13-s − 0.213·16-s + (0.196 − 0.340i)17-s + (−1.61 + 0.932i)19-s + (0.217 − 0.377i)20-s + (−0.427 − 0.740i)22-s + (0.778 + 0.449i)23-s + (0.146 + 0.253i)25-s + (−0.390 − 0.676i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.282 + 0.959i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.282 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.327249479\)
\(L(\frac12)\) \(\approx\) \(2.327249479\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.981iT - 2T^{2} \)
5 \( 1 + (-0.940 + 1.62i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.54 + 2.04i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.51 + 2.02i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.810 + 1.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.03 - 4.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.73 - 2.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.542 - 0.313i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.27iT - 31T^{2} \)
37 \( 1 + (3.97 + 6.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.912 + 1.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.53 - 6.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.93T + 47T^{2} \)
53 \( 1 + (-7.24 - 4.18i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.17T + 59T^{2} \)
61 \( 1 + 3.74iT - 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + (-3.28 - 1.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 8.36T + 79T^{2} \)
83 \( 1 + (4.38 - 7.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.90 + 8.49i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.4 - 6.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.354977429250132787950319956374, −8.806746094827743058785429614694, −7.934148352683679709335070476945, −6.70170733358340776080793661647, −6.16127057368969609724433231469, −5.19185816614550280108878087865, −3.89702083756256987197202823039, −3.24916292555792393094547615806, −1.82396282255830445906770831372, −1.04225815382723315226100752628, 1.68094197948728875558023645304, 2.61184506556384665195907887033, 3.88395883799872034043386489054, 4.92299330243359128706348536041, 6.17104959912880435744422463576, 6.64447683420575329319334496750, 6.98268487593331994494108031579, 8.323900356552833354709997922945, 8.797395776946520805895166374890, 9.897570358010782025406672690511

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