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Properties

Label 2-1323-63.38-c1-0-33
Degree $2$
Conductor $1323$
Sign $0.0262 + 0.999i$
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  − 0.424i·2-s + 1.82·4-s + (1.80 − 3.12i)5-s − 1.62i·8-s + (−1.32 − 0.765i)10-s + (3.20 − 1.85i)11-s + (−5.23 + 3.02i)13-s + 2.95·16-s + (0.532 − 0.921i)17-s + (3.16 − 1.82i)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 + (1.28 + 2.22i)26-s + ⋯
L(s)  = 1  − 0.299i·2-s + 0.910·4-s + (0.806 − 1.39i)5-s − 0.572i·8-s + (−0.419 − 0.241i)10-s + (0.967 − 0.558i)11-s + (−1.45 + 0.838i)13-s + 0.738·16-s + (0.129 − 0.223i)17-s + (0.725 − 0.418i)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 + (0.251 + 0.435i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0262 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0262 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.0262 + 0.999i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.0262 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.445906207\)
\(L(\frac12)\) \(\approx\) \(2.445906207\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.424iT - 2T^{2} \)
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 \( 1 + (-0.532 + 0.921i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.16 + 1.82i)T + (9.5 - 16.4i)T^{2} \)
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 \( 1 + 1.08iT - 31T^{2} \)
37 \( 1 + (-4.00 - 6.93i)T + (-18.5 + 32.0i)T^{2} \)
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 \( 1 + 5.67T + 47T^{2} \)
53 \( 1 + (-3.92 - 2.26i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 0.0275iT - 61T^{2} \)
67 \( 1 + 9.72T + 67T^{2} \)
71 \( 1 - 5.55iT - 71T^{2} \)
73 \( 1 + (-1.95 - 1.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 6.53T + 79T^{2} \)
83 \( 1 + (-1.52 + 2.64i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.47 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
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

−9.518241288087881578160720348665, −8.870111602944467907620519110346, −7.80864009934606370850160949679, −6.87974933924508931662247872052, −6.13054843936784061808453120918, −5.17961808114228058653614494561, −4.40609970293063715022739103517, −3.04860974000726252582874555405, −1.93316609497012288980117056410, −1.04153329817888150414148476672, 1.76633291678743352292161931769, 2.64069165420284163169361691540, 3.43836537363729383970010059513, 5.00697300179130321059454899065, 5.96347457111317029071689372229, 6.54903171519203002971832016638, 7.34452509785288264161839264344, 7.75085724420012726398753241011, 9.224379630205654714089794813016, 10.07139921287360683043288000014

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