Properties

Label 2-1323-63.20-c1-0-14
Degree $2$
Conductor $1323$
Sign $-0.687 - 0.725i$
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  + (2.05 + 1.18i)2-s + (1.81 + 3.14i)4-s + (1.71 + 2.97i)5-s + 3.86i·8-s + 8.15i·10-s + (−0.271 − 0.156i)11-s + (−5.09 + 2.94i)13-s + (−0.958 + 1.65i)16-s − 0.953·17-s + 1.26i·19-s + (−6.23 + 10.7i)20-s + (−0.372 − 0.645i)22-s + (5.91 − 3.41i)23-s + (−3.40 + 5.89i)25-s − 13.9·26-s + ⋯
L(s)  = 1  + (1.45 + 0.838i)2-s + (0.907 + 1.57i)4-s + (0.768 + 1.33i)5-s + 1.36i·8-s + 2.57i·10-s + (−0.0819 − 0.0473i)11-s + (−1.41 + 0.816i)13-s + (−0.239 + 0.414i)16-s − 0.231·17-s + 0.289i·19-s + (−1.39 + 2.41i)20-s + (−0.0794 − 0.137i)22-s + (1.23 − 0.711i)23-s + (−0.680 + 1.17i)25-s − 2.73·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.687 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.687 - 0.725i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.687 - 0.725i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.013177178\)
\(L(\frac12)\) \(\approx\) \(4.013177178\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.05 - 1.18i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.71 - 2.97i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.271 + 0.156i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.09 - 2.94i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.953T + 17T^{2} \)
19 \( 1 - 1.26iT - 19T^{2} \)
23 \( 1 + (-5.91 + 3.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.43 + 1.98i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.53 + 2.61i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.37T + 37T^{2} \)
41 \( 1 + (0.0699 + 0.121i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.44 + 2.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.00 + 1.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.9iT - 53T^{2} \)
59 \( 1 + (0.824 + 1.42i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.57 - 1.48i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.934 - 1.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + 0.409iT - 73T^{2} \)
79 \( 1 + (5.23 - 9.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.00 + 6.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.11T + 89T^{2} \)
97 \( 1 + (-10.5 - 6.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.969096398494969043661023813656, −9.188424017864028942059592380549, −7.76350204762909869590372364668, −7.11669652962669453592801473330, −6.54071825941760600859271690588, −5.87360661957207789382391795554, −4.93234500323802972309900772526, −4.12557890223096058343474071718, −2.91122298053212432538874929884, −2.34331090304490345572842333216, 1.05682934131158522749902310542, 2.21129877241852080150135333844, 3.08895190648553174944754794547, 4.35229027115196299039471692150, 5.17824105565700476389681628994, 5.32081583082239685804967738864, 6.47375295406229386808921953068, 7.62943847494210591046673239008, 8.735797787459677271789949413473, 9.595332882818485513610095225085

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