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Properties

Label 2-1323-9.4-c1-0-21
Degree $2$
Conductor $1323$
Sign $-0.999 - 0.0431i$
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  + (−1.35 − 2.35i)2-s + (−2.68 + 4.65i)4-s + (0.793 − 1.37i)5-s + 9.15·8-s − 4.30·10-s + (−0.674 − 1.16i)11-s + (−1.58 + 2.75i)13-s + (−7.05 − 12.2i)16-s + 2.80·17-s − 0.625·19-s + (4.26 + 7.38i)20-s + (−1.83 + 3.17i)22-s + (−0.142 + 0.246i)23-s + (1.24 + 2.15i)25-s + 8.62·26-s + ⋯
L(s)  = 1  + (−0.959 − 1.66i)2-s + (−1.34 + 2.32i)4-s + (0.354 − 0.614i)5-s + 3.23·8-s − 1.36·10-s + (−0.203 − 0.352i)11-s + (−0.440 + 0.763i)13-s + (−1.76 − 3.05i)16-s + 0.679·17-s − 0.143·19-s + (0.952 + 1.65i)20-s + (−0.390 + 0.676i)22-s + (−0.0296 + 0.0514i)23-s + (0.248 + 0.430i)25-s + 1.69·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.999 - 0.0431i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.999 - 0.0431i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7714163624\)
\(L(\frac12)\) \(\approx\) \(0.7714163624\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 + 2.35i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.793 + 1.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.674 + 1.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.58 - 2.75i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.80T + 17T^{2} \)
19 \( 1 + 0.625T + 19T^{2} \)
23 \( 1 + (0.142 - 0.246i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.27 + 3.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.71 + 6.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.02T + 37T^{2} \)
41 \( 1 + (-5.01 + 8.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.12 + 5.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.57 + 9.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.78T + 53T^{2} \)
59 \( 1 + (2.28 - 3.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.192 + 0.333i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.26 + 2.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.45T + 71T^{2} \)
73 \( 1 - 0.468T + 73T^{2} \)
79 \( 1 + (-7.85 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.99 + 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.58T + 89T^{2} \)
97 \( 1 + (7.22 + 12.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.548252658506712086350317246825, −8.689954658320365357889195552325, −8.033252813804817960875409477841, −7.16473057309031270439810467887, −5.67530313886878991601999539073, −4.56606915491264718180698039590, −3.74869657141030485213010146625, −2.60974126020302696466048406530, −1.72946753633645133644770257536, −0.51333337749616505730899448014, 1.19991907360906866995408812626, 2.86047633590824629154337186491, 4.59062601120360958243632232205, 5.31587420152606957653582449281, 6.26775992438528532827030424235, 6.72492089671696979719018988828, 7.83611190209469751182956556058, 7.992765518238916945774601483668, 9.204480822004508976737993045019, 9.803441872784334480443492972449

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