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Properties

Label 2-1323-63.58-c1-0-33
Degree $2$
Conductor $1323$
Sign $-0.0538 + 0.998i$
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  + 1.69·2-s + 0.888·4-s + (−0.474 − 0.822i)5-s − 1.88·8-s + (−0.806 − 1.39i)10-s + (−0.294 + 0.509i)11-s + (2.50 − 4.34i)13-s − 4.98·16-s + (−3.79 − 6.56i)17-s + (2.23 − 3.86i)19-s + (−0.421 − 0.730i)20-s + (−0.5 + 0.866i)22-s + (1.23 + 2.14i)23-s + (2.04 − 3.54i)25-s + (4.26 − 7.38i)26-s + ⋯
L(s)  = 1  + 1.20·2-s + 0.444·4-s + (−0.212 − 0.367i)5-s − 0.667·8-s + (−0.255 − 0.441i)10-s + (−0.0886 + 0.153i)11-s + (0.696 − 1.20i)13-s − 1.24·16-s + (−0.919 − 1.59i)17-s + (0.511 − 0.886i)19-s + (−0.0943 − 0.163i)20-s + (−0.106 + 0.184i)22-s + (0.258 + 0.447i)23-s + (0.409 − 0.709i)25-s + (0.836 − 1.44i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0538 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0538 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.0538 + 0.998i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.0538 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.193997409\)
\(L(\frac12)\) \(\approx\) \(2.193997409\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.69T + 2T^{2} \)
5 \( 1 + (0.474 + 0.822i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.294 - 0.509i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.50 + 4.34i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.79 + 6.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.23 + 3.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.23 - 2.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.73 - 4.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.07T + 31T^{2} \)
37 \( 1 + (-3.49 + 6.05i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.527 - 0.913i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.49 + 6.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.47T + 47T^{2} \)
53 \( 1 + (-3.46 - 5.99i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 + 4.30T + 71T^{2} \)
73 \( 1 + (2.23 + 3.86i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 + (2.84 + 4.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.421 + 0.730i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.70 - 2.94i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.067788522514340865211610290847, −8.921838605535840192007054132990, −7.57395493495847382123586537325, −6.85204725616969424474764697661, −5.75582321078820903096886231542, −5.10653194103196350530087062352, −4.42421132870510771231274328289, −3.34570063312164063292503750511, −2.58131740208272035024608110006, −0.60556554721176308445564204384, 1.76719316640938997915984305179, 3.07908913615572750064050538736, 3.95656149219504402464756954029, 4.50336173874167921114045368950, 5.72805747268866245537802747738, 6.30497343997637880137507008198, 7.06519670374042906367128721506, 8.311432265235887330938814834354, 8.907543492513571493093718614782, 9.897338712768152180017525630662

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