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Properties

Label 2-1323-63.41-c1-0-9
Degree $2$
Conductor $1323$
Sign $-0.501 - 0.864i$
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  + (−2.24 + 1.29i)2-s + (2.36 − 4.09i)4-s + (−0.626 + 1.08i)5-s + 7.07i·8-s − 3.24i·10-s + (−0.534 + 0.308i)11-s + (1.06 + 0.613i)13-s + (−4.44 − 7.69i)16-s + 4.43·17-s + 1.90i·19-s + (2.96 + 5.12i)20-s + (0.799 − 1.38i)22-s + (−4.11 − 2.37i)23-s + (1.71 + 2.97i)25-s − 3.18·26-s + ⋯
L(s)  = 1  + (−1.58 + 0.916i)2-s + (1.18 − 2.04i)4-s + (−0.280 + 0.485i)5-s + 2.50i·8-s − 1.02i·10-s + (−0.161 + 0.0929i)11-s + (0.294 + 0.170i)13-s + (−1.11 − 1.92i)16-s + 1.07·17-s + 0.436i·19-s + (0.662 + 1.14i)20-s + (0.170 − 0.295i)22-s + (−0.857 − 0.495i)23-s + (0.343 + 0.594i)25-s − 0.624·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.501 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.501 - 0.864i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.501 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6195683944\)
\(L(\frac12)\) \(\approx\) \(0.6195683944\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.24 - 1.29i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (0.626 - 1.08i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.534 - 0.308i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.06 - 0.613i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.43T + 17T^{2} \)
19 \( 1 - 1.90iT - 19T^{2} \)
23 \( 1 + (4.11 + 2.37i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.07 + 2.93i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.14 + 1.24i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.66T + 37T^{2} \)
41 \( 1 + (-2.09 + 3.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.24 + 3.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.80 - 6.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.09iT - 53T^{2} \)
59 \( 1 + (-1.78 + 3.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12.5 + 7.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.80 - 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 - 11.4iT - 73T^{2} \)
79 \( 1 + (-2.01 - 3.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.36 - 7.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.62T + 89T^{2} \)
97 \( 1 + (-8.76 + 5.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.912048295771503500128706969241, −8.923987301341477736711447862521, −8.233346423752361782111608564108, −7.58191361097430904305988544073, −6.89856717424518901223288664802, −6.08308567850782925076022031874, −5.31761572308185820735387737979, −3.81182835772548776822985390728, −2.37338275827050266595700232326, −1.03731346991581652496362093993, 0.55198153948065139759623053917, 1.64240301211239038508646758404, 2.87898240359564145891895896251, 3.73430324435494573549746325548, 5.03207656428126614738171122268, 6.32724678743228778162694655001, 7.39786244359138342833540925053, 8.035712548626290378138713516921, 8.670294090028806715142242652807, 9.340761075172646799642405836590

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