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Properties

Label 2-1323-63.20-c1-0-12
Degree $2$
Conductor $1323$
Sign $0.293 + 0.955i$
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  + (−0.575 − 0.332i)2-s + (−0.779 − 1.34i)4-s + (−0.0141 − 0.0245i)5-s + 2.36i·8-s + 0.0188i·10-s + (0.885 + 0.511i)11-s + (−4.87 + 2.81i)13-s + (−0.773 + 1.33i)16-s + 5.67·17-s − 2.09i·19-s + (−0.0220 + 0.0382i)20-s + (−0.339 − 0.588i)22-s + (6.28 − 3.63i)23-s + (2.49 − 4.32i)25-s + 3.74·26-s + ⋯
L(s)  = 1  + (−0.406 − 0.234i)2-s + (−0.389 − 0.674i)4-s + (−0.00632 − 0.0109i)5-s + 0.835i·8-s + 0.00594i·10-s + (0.266 + 0.154i)11-s + (−1.35 + 0.781i)13-s + (−0.193 + 0.334i)16-s + 1.37·17-s − 0.480i·19-s + (−0.00493 + 0.00854i)20-s + (−0.0723 − 0.125i)22-s + (1.31 − 0.757i)23-s + (0.499 − 0.865i)25-s + 0.733·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.293 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.293 + 0.955i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.293 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.085398564\)
\(L(\frac12)\) \(\approx\) \(1.085398564\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.575 + 0.332i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.0141 + 0.0245i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.885 - 0.511i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.87 - 2.81i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.67T + 17T^{2} \)
19 \( 1 + 2.09iT - 19T^{2} \)
23 \( 1 + (-6.28 + 3.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.52 - 2.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.87 + 1.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.47T + 37T^{2} \)
41 \( 1 + (3.52 + 6.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.15 - 2.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.43 - 9.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 + (-3.01 - 5.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.05 - 1.18i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.38 + 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.93iT - 71T^{2} \)
73 \( 1 + 10.8iT - 73T^{2} \)
79 \( 1 + (-7.80 + 13.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.07 - 5.32i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (-6.77 - 3.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.474720909354441220800643252747, −8.897885176819343986454845334975, −7.984078028765797208041497791807, −6.99974807729796674697213213113, −6.22633633192463014535957722886, −4.93221795368392543278017347763, −4.71909926504863329509431999706, −3.10684150577201917512729872316, −1.98101622032724741976496463421, −0.67326252360322110116251968501, 1.03189598797801917096639150261, 2.89376278182216018561902670579, 3.53710489627293480376239770791, 4.82022133197944447452577908464, 5.50262596432442833402904073163, 6.84768664640603513614600980470, 7.44042194372821395676613353311, 8.149094921131390226485501975218, 8.898794885293726690290795686730, 9.858598729954816245574697399560

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