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Properties

Label 2-1323-63.41-c1-0-6
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: $\chi_{1323} (881, \cdot )$
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

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

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