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Properties

Label 2-1323-63.41-c1-0-0
Degree $2$
Conductor $1323$
Sign $-0.999 + 0.0311i$
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  + (−1.80 + 1.04i)2-s + (1.17 − 2.03i)4-s + (1.65 − 2.86i)5-s + 0.717i·8-s + 6.88i·10-s + (−2.30 + 1.33i)11-s + (−2.11 − 1.21i)13-s + (1.59 + 2.76i)16-s − 7.18·17-s − 4.90i·19-s + (−3.87 − 6.70i)20-s + (2.77 − 4.80i)22-s + (4.32 + 2.49i)23-s + (−2.96 − 5.12i)25-s + 5.08·26-s + ⋯
L(s)  = 1  + (−1.27 + 0.736i)2-s + (0.586 − 1.01i)4-s + (0.738 − 1.27i)5-s + 0.253i·8-s + 2.17i·10-s + (−0.694 + 0.401i)11-s + (−0.585 − 0.338i)13-s + (0.399 + 0.691i)16-s − 1.74·17-s − 1.12i·19-s + (−0.866 − 1.50i)20-s + (0.591 − 1.02i)22-s + (0.901 + 0.520i)23-s + (−0.592 − 1.02i)25-s + 0.997·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.999 + 0.0311i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.999 + 0.0311i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.009462952453\)
\(L(\frac12)\) \(\approx\) \(0.009462952453\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 - 1.04i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.65 + 2.86i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.30 - 1.33i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.11 + 1.21i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.18T + 17T^{2} \)
19 \( 1 + 4.90iT - 19T^{2} \)
23 \( 1 + (-4.32 - 2.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.50 - 3.17i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.30 + 1.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.68T + 37T^{2} \)
41 \( 1 + (-0.553 + 0.958i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.93 - 5.08i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.44 - 4.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + (2.56 - 4.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.44 - 2.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.16 - 7.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.07iT - 71T^{2} \)
73 \( 1 - 8.01iT - 73T^{2} \)
79 \( 1 + (2.50 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.04 - 1.80i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.08T + 89T^{2} \)
97 \( 1 + (-9.47 + 5.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.579828843590600014865025006141, −9.061030327933666405836206882993, −8.748801608543519406042264787114, −7.59675118774613298121149984751, −7.10475448713415517119271285216, −6.02171702483192308368103076643, −5.17394172039647490654602645329, −4.41241005241293276897593323442, −2.53198644977878426417449855704, −1.33643750900251354436440273208, 0.00589164195832252731494637731, 1.96648270748726042445413042259, 2.45556501791949226692508881603, 3.52060285533209208798065636595, 5.05554743191158706590785467283, 6.13150237321374550217229269756, 6.97215450159192683683991085092, 7.71538381978128346455944074566, 8.688002747234000358107444366604, 9.359393027279586574670590678667

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