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Properties

Label 2-1323-63.38-c1-0-19
Degree $2$
Conductor $1323$
Sign $-0.151 - 0.988i$
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  + 1.83i·2-s − 1.35·4-s + (−0.322 + 0.559i)5-s + 1.17i·8-s + (−1.02 − 0.591i)10-s + (4.60 − 2.65i)11-s + (4.44 − 2.56i)13-s − 4.87·16-s + (0.814 − 1.41i)17-s + (2.09 − 1.20i)19-s + (0.437 − 0.758i)20-s + (4.86 + 8.43i)22-s + (−1.27 − 0.735i)23-s + (2.29 + 3.96i)25-s + (4.69 + 8.13i)26-s + ⋯
L(s)  = 1  + 1.29i·2-s − 0.678·4-s + (−0.144 + 0.250i)5-s + 0.416i·8-s + (−0.323 − 0.187i)10-s + (1.38 − 0.801i)11-s + (1.23 − 0.711i)13-s − 1.21·16-s + (0.197 − 0.342i)17-s + (0.479 − 0.276i)19-s + (0.0978 − 0.169i)20-s + (1.03 + 1.79i)22-s + (−0.265 − 0.153i)23-s + (0.458 + 0.793i)25-s + (0.921 + 1.59i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.151 - 0.988i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.151 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.957491355\)
\(L(\frac12)\) \(\approx\) \(1.957491355\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.83iT - 2T^{2} \)
5 \( 1 + (0.322 - 0.559i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.60 + 2.65i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.44 + 2.56i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.814 + 1.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.09 + 1.20i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.27 + 0.735i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.43 - 3.71i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.66iT - 31T^{2} \)
37 \( 1 + (-3.99 - 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.99 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.51 - 2.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.08T + 47T^{2} \)
53 \( 1 + (2.04 + 1.18i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.95T + 59T^{2} \)
61 \( 1 - 10.6iT - 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 - 4.76iT - 71T^{2} \)
73 \( 1 + (-10.2 - 5.90i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 6.96T + 79T^{2} \)
83 \( 1 + (3.51 - 6.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.16 - 3.74i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.3 + 8.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.553574887792354621757411340198, −8.660119998198819683807266449758, −8.278417746285805186377529986222, −7.23717671615054979857311023339, −6.56220720649333360824029690718, −5.94293405961392152782864859158, −5.11388958923728950202789792410, −3.90777952386672021595062383319, −2.97276242092956357039066892293, −1.14442106591776850020353064706, 1.12002580831837932922236671966, 1.87111306816423497471940535333, 3.24542063659600497110493133845, 4.02667145453624292358035075603, 4.70744465167300415085992827153, 6.31469466717628080528162760746, 6.71268025463038232163093677879, 8.029274906195883005947244622578, 8.922129682578585923065903329129, 9.545968949282212374525047390149

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