Properties

Label 2-1323-63.20-c1-0-33
Degree $2$
Conductor $1323$
Sign $-0.985 - 0.170i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.985 - 0.170i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.985 - 0.170i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8749219622\)
\(L(\frac12)\) \(\approx\) \(0.8749219622\)
\(L(\frac{3}{2})\) 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

−9.014545143898559936638895785329, −8.413937800142930721876098305063, −7.88839642482057795179492968043, −6.54152014965670303906725257943, −5.54793294258254183137148895326, −5.08837021825470064800662659654, −4.19777768311497191260809406345, −3.23474891277576184180963748989, −1.28829622732123936175949070868, −0.36825643789052549301119644112, 2.15049613299976153479859144020, 3.32786954860632644841279780869, 3.74777258278444185646803214811, 4.75810315037019137311171496983, 5.99598835287446189554156728950, 6.95583301906834277846730992279, 7.63118585530680685496591559110, 8.277749608261875085379592521605, 9.120114395639277983308676246658, 10.39106984733238877074784684927

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