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Properties

Label 2-1323-9.4-c1-0-26
Degree $2$
Conductor $1323$
Sign $0.492 + 0.870i$
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  + (0.119 + 0.207i)2-s + (0.971 − 1.68i)4-s + (1.29 − 2.24i)5-s + 0.942·8-s + 0.619·10-s + (2.09 + 3.62i)11-s + (1.84 − 3.18i)13-s + (−1.83 − 3.16i)16-s + 1.71·17-s + 7.15·19-s + (−2.51 − 4.36i)20-s + (−0.500 + 0.866i)22-s + (−2.56 + 4.43i)23-s + (−0.858 − 1.48i)25-s + 0.880·26-s + ⋯
L(s)  = 1  + (0.0845 + 0.146i)2-s + (0.485 − 0.841i)4-s + (0.579 − 1.00i)5-s + 0.333·8-s + 0.195·10-s + (0.630 + 1.09i)11-s + (0.510 − 0.884i)13-s + (−0.457 − 0.792i)16-s + 0.414·17-s + 1.64·19-s + (−0.562 − 0.975i)20-s + (−0.106 + 0.184i)22-s + (−0.534 + 0.925i)23-s + (−0.171 − 0.297i)25-s + 0.172·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.492 + 0.870i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.492 + 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.401832953\)
\(L(\frac12)\) \(\approx\) \(2.401832953\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.119 - 0.207i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.29 + 2.24i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.09 - 3.62i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.84 + 3.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.71T + 17T^{2} \)
19 \( 1 - 7.15T + 19T^{2} \)
23 \( 1 + (2.56 - 4.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.06 + 1.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.26 - 5.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.66T + 37T^{2} \)
41 \( 1 + (-5.10 + 8.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.830 - 1.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.66 + 8.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 + (3.03 - 5.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.99 + 6.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.13 - 7.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.23T + 71T^{2} \)
73 \( 1 + 7.15T + 73T^{2} \)
79 \( 1 + (-4.91 - 8.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.44 + 5.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.03T + 89T^{2} \)
97 \( 1 + (-1.53 - 2.65i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.621384964523199505990733972941, −8.910284075433022361357184340974, −7.69582576224342059159470654044, −7.08631938719600098723822266768, −5.87849001124439037669728477917, −5.46023454038874809992165201446, −4.65958027591032543612782159491, −3.33924639120620900101923733238, −1.81221675905534425231295381681, −1.11289575825172498147000454340, 1.55491300754582599388401451211, 2.83609422897789860176796927309, 3.39020113295094286146593754025, 4.44057220095035660432042063336, 6.03613423552338698019934248815, 6.32370533683515609412718768990, 7.32978418179376396594643304527, 8.008466509493514273171559721765, 9.034366208660635162317722703317, 9.729320221581559922491259842649

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