Properties

Label 2-1323-63.4-c1-0-12
Degree $2$
Conductor $1323$
Sign $0.394 - 0.919i$
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 + 2.59·5-s + 0.942·8-s + (0.309 − 0.536i)10-s − 4.18·11-s + (−1.84 + 3.18i)13-s + (−1.83 + 3.16i)16-s + (0.855 − 1.48i)17-s + (3.57 + 6.19i)19-s + (2.51 + 4.36i)20-s + (−0.499 + 0.866i)22-s + 5.12·23-s + 1.71·25-s + (0.440 + 0.762i)26-s + ⋯
L(s)  = 1  + (0.0845 − 0.146i)2-s + (0.485 + 0.841i)4-s + 1.15·5-s + 0.333·8-s + (0.0979 − 0.169i)10-s − 1.26·11-s + (−0.510 + 0.884i)13-s + (−0.457 + 0.792i)16-s + (0.207 − 0.359i)17-s + (0.820 + 1.42i)19-s + (0.562 + 0.975i)20-s + (−0.106 + 0.184i)22-s + 1.06·23-s + 0.343·25-s + (0.0863 + 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.394 - 0.919i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.394 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.203072267\)
\(L(\frac12)\) \(\approx\) \(2.203072267\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.59T + 5T^{2} \)
11 \( 1 + 4.18T + 11T^{2} \)
13 \( 1 + (1.84 - 3.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.855 + 1.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.57 - 6.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.12T + 23T^{2} \)
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 \( 1 + (0.830 + 1.43i)T + (-18.5 + 32.0i)T^{2} \)
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 \( 1 + (-5.32 + 9.22i)T + (-26.5 - 45.8i)T^{2} \)
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 \( 1 + (3.57 - 6.19i)T + (-36.5 - 63.2i)T^{2} \)
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 \( 1 + (2.51 + 4.36i)T + (-44.5 + 77.0i)T^{2} \)
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

−9.975352744375175042204702101185, −8.993364082761331689508559696108, −8.095264702106197470275971071813, −7.33333248899498786882541659326, −6.57168459880016558721629007504, −5.55100224751857458245569097347, −4.81086665852899545704645874742, −3.46045355546798776187721945798, −2.57913835387543878902828257225, −1.69568617851227494755837155947, 0.870409223148638665232873813034, 2.28474505584235212381072481845, 2.89942236397684887495704802629, 4.79013565417520332287376365555, 5.44152088307267919857078606045, 5.90581160126552098881566737568, 7.04627509557623198147241369173, 7.59598871910710869978371548285, 8.864831538988767627386949528938, 9.638628170883709943683794467931

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