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Properties

Label 2-1323-63.41-c1-0-32
Degree $2$
Conductor $1323$
Sign $-0.884 + 0.467i$
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  + (0.367 − 0.212i)2-s + (−0.910 + 1.57i)4-s + (1.80 − 3.12i)5-s + 1.62i·8-s − 1.53i·10-s + (−3.20 + 1.85i)11-s + (−5.23 − 3.02i)13-s + (−1.47 − 2.55i)16-s − 1.06·17-s − 3.65i·19-s + (3.28 + 5.68i)20-s + (−0.786 + 1.36i)22-s + (−0.314 − 0.181i)23-s + (−4.00 − 6.94i)25-s − 2.56·26-s + ⋯
L(s)  = 1  + (0.259 − 0.149i)2-s + (−0.455 + 0.788i)4-s + (0.806 − 1.39i)5-s + 0.572i·8-s − 0.483i·10-s + (−0.967 + 0.558i)11-s + (−1.45 − 0.838i)13-s + (−0.369 − 0.639i)16-s − 0.258·17-s − 0.837i·19-s + (0.734 + 1.27i)20-s + (−0.167 + 0.290i)22-s + (−0.0655 − 0.0378i)23-s + (−0.801 − 1.38i)25-s − 0.502·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.884 + 0.467i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.884 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6886023793\)
\(L(\frac12)\) \(\approx\) \(0.6886023793\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.367 + 0.212i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.80 + 3.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.20 - 1.85i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.23 + 3.02i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.06T + 17T^{2} \)
19 \( 1 + 3.65iT - 19T^{2} \)
23 \( 1 + (0.314 + 0.181i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.857 + 0.495i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.939 + 0.542i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.00T + 37T^{2} \)
41 \( 1 + (2.09 - 3.62i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.89 + 3.28i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.83 - 4.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.53iT - 53T^{2} \)
59 \( 1 + (-5.62 + 9.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0238 - 0.0137i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.86 + 8.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.55iT - 71T^{2} \)
73 \( 1 - 2.25iT - 73T^{2} \)
79 \( 1 + (3.26 + 5.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.52 - 2.64i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.9T + 89T^{2} \)
97 \( 1 + (1.67 - 0.964i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.338716161581285760449251357315, −8.471859849029899984727054091334, −7.85588670217906333418172742225, −6.97127158056518062030414092690, −5.44057265899194360768805810781, −5.04802303226006739454920774399, −4.40956313819086664831999033687, −2.94252900267668303322220525305, −2.06675339970135778257895061265, −0.23853089116607163417054540659, 1.88772103022158191635658852950, 2.77757889024767288033672102071, 3.98544222133507170880191026052, 5.21440029197988608079691686401, 5.70022065757535615809362822031, 6.73563785429623638632065114264, 7.14828747617887550121741266505, 8.408627380554251755559190010978, 9.429146956757255687119239467200, 10.15697929920969897286877639621

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