Properties

Label 2-1323-63.41-c1-0-21
Degree $2$
Conductor $1323$
Sign $0.974 - 0.224i$
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  + (−2.23 + 1.28i)2-s + (2.32 − 4.02i)4-s + (1.16 − 2.01i)5-s + 6.82i·8-s + 6.01i·10-s + (3.78 − 2.18i)11-s + (1.14 + 0.660i)13-s + (−4.15 − 7.18i)16-s + 5.78·17-s + 0.675i·19-s + (−5.41 − 9.38i)20-s + (−5.62 + 9.74i)22-s + (4.81 + 2.78i)23-s + (−0.218 − 0.379i)25-s − 3.40·26-s + ⋯
L(s)  = 1  + (−1.57 + 0.911i)2-s + (1.16 − 2.01i)4-s + (0.521 − 0.903i)5-s + 2.41i·8-s + 1.90i·10-s + (1.14 − 0.658i)11-s + (0.317 + 0.183i)13-s + (−1.03 − 1.79i)16-s + 1.40·17-s + 0.154i·19-s + (−1.21 − 2.09i)20-s + (−1.19 + 2.07i)22-s + (1.00 + 0.580i)23-s + (−0.0437 − 0.0758i)25-s − 0.667·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.974 - 0.224i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.974 - 0.224i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9843898623\)
\(L(\frac12)\) \(\approx\) \(0.9843898623\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 1.28i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.16 + 2.01i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.78 + 2.18i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.14 - 0.660i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.78T + 17T^{2} \)
19 \( 1 - 0.675iT - 19T^{2} \)
23 \( 1 + (-4.81 - 2.78i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.86 - 2.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.47 + 2.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.01T + 37T^{2} \)
41 \( 1 + (3.29 - 5.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.89 - 6.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.246 - 0.427i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.14iT - 53T^{2} \)
59 \( 1 + (-2.15 + 3.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.77 - 1.02i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.41 + 4.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.17iT - 71T^{2} \)
73 \( 1 + 15.1iT - 73T^{2} \)
79 \( 1 + (-5.30 - 9.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.32 + 9.22i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.32T + 89T^{2} \)
97 \( 1 + (-12.7 + 7.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.239952959001593035317190319791, −9.082359155630206714122418810625, −8.124446292847363752382784694105, −7.45255798935528380077735572548, −6.45073123912475306130608431096, −5.81622838215312194114879783125, −5.04523611177663634928768326421, −3.47172643751591616172759991628, −1.59735527008900081803402444701, −0.968786025469877318273834721072, 1.04621369071472486997298962326, 2.11327199050647779839314791699, 3.05456182964738755375660752980, 3.94703431860264740860700495787, 5.63831132659839623810612484821, 6.82257617841083938353391155062, 7.21981508952982840403844219053, 8.212063084220555100351802070701, 9.112759809457833908372469776796, 9.589984566176349667173303491643

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