Properties

Label 2-1323-63.58-c1-0-15
Degree $2$
Conductor $1323$
Sign $0.975 - 0.220i$
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  + 0.879·2-s − 1.22·4-s + (0.673 + 1.16i)5-s − 2.83·8-s + (0.592 + 1.02i)10-s + (0.826 − 1.43i)11-s + (1.68 − 2.91i)13-s − 0.0418·16-s + (0.233 + 0.405i)17-s + (1.61 − 2.79i)19-s + (−0.826 − 1.43i)20-s + (0.726 − 1.25i)22-s + (4.47 + 7.74i)23-s + (1.59 − 2.75i)25-s + (1.48 − 2.56i)26-s + ⋯
L(s)  = 1  + 0.621·2-s − 0.613·4-s + (0.301 + 0.521i)5-s − 1.00·8-s + (0.187 + 0.324i)10-s + (0.249 − 0.431i)11-s + (0.467 − 0.809i)13-s − 0.0104·16-s + (0.0567 + 0.0982i)17-s + (0.370 − 0.641i)19-s + (−0.184 − 0.320i)20-s + (0.154 − 0.268i)22-s + (0.932 + 1.61i)23-s + (0.318 − 0.551i)25-s + (0.290 − 0.503i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.975 - 0.220i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.029574706\)
\(L(\frac12)\) \(\approx\) \(2.029574706\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.879T + 2T^{2} \)
5 \( 1 + (-0.673 - 1.16i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.826 + 1.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.68 + 2.91i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.233 - 0.405i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.61 + 2.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.47 - 7.74i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.13 - 5.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.23T + 31T^{2} \)
37 \( 1 + (4.61 - 7.99i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.70 + 2.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.20 - 3.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.35T + 47T^{2} \)
53 \( 1 + (0.286 + 0.497i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 7.63T + 61T^{2} \)
67 \( 1 - 0.596T + 67T^{2} \)
71 \( 1 - 0.554T + 71T^{2} \)
73 \( 1 + (1.02 + 1.77i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 2.40T + 79T^{2} \)
83 \( 1 + (7.52 + 13.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.54 + 7.86i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.949 - 1.64i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.735736950274807242377486425071, −8.782734489746828852664306773207, −8.213179536753930382631998893451, −6.99447707426703190656588965029, −6.22694358266150115256101023063, −5.37160932839038192665285468334, −4.67000187049022545446688180544, −3.37981372863964212290991161340, −2.95406513760523355890324987096, −1.03744470626005648041139332905, 0.974039639536031322478803260247, 2.49642580341731286473168079976, 3.75782377640479351629457596012, 4.53272063276787728897164247088, 5.20399056228487352996145724472, 6.18223084513904871590622588065, 6.90801745634069651221251159207, 8.248369554365298655138041841391, 8.781953651579833933758631414120, 9.552542562088576796130744768628

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