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Properties

Label 2-1323-63.25-c1-0-21
Degree $2$
Conductor $1323$
Sign $0.501 - 0.864i$
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  + 2.46·2-s + 4.05·4-s + (−1.29 + 2.24i)5-s + 5.05·8-s + (−3.19 + 5.52i)10-s + (2.25 + 3.90i)11-s + (−0.5 − 0.866i)13-s + 4.32·16-s + (−0.472 + 0.819i)17-s + (2.02 + 3.51i)19-s + (−5.25 + 9.10i)20-s + (5.55 + 9.61i)22-s + (−0.136 + 0.236i)23-s + (−0.863 − 1.49i)25-s + (−1.23 − 2.13i)26-s + ⋯
L(s)  = 1  + 1.73·2-s + 2.02·4-s + (−0.579 + 1.00i)5-s + 1.78·8-s + (−1.00 + 1.74i)10-s + (0.680 + 1.17i)11-s + (−0.138 − 0.240i)13-s + 1.08·16-s + (−0.114 + 0.198i)17-s + (0.465 + 0.805i)19-s + (−1.17 + 2.03i)20-s + (1.18 + 2.05i)22-s + (−0.0284 + 0.0493i)23-s + (−0.172 − 0.299i)25-s + (−0.241 − 0.417i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.501 - 0.864i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.501 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.275009263\)
\(L(\frac12)\) \(\approx\) \(4.275009263\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.46T + 2T^{2} \)
5 \( 1 + (1.29 - 2.24i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.25 - 3.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.472 - 0.819i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.02 - 3.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.136 - 0.236i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.23 + 2.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.32T + 31T^{2} \)
37 \( 1 + (0.890 + 1.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.20 + 5.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.21 + 9.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + (3.13 - 5.43i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.72T + 59T^{2} \)
61 \( 1 + 2.27T + 61T^{2} \)
67 \( 1 + 15.8T + 67T^{2} \)
71 \( 1 + 3.27T + 71T^{2} \)
73 \( 1 + (-0.753 + 1.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + (0.472 - 0.819i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.17 + 12.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.74 + 9.95i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.16529776354242139436793793817, −8.963118255324652020929103084499, −7.48583356560046007316880757135, −7.23180306883817513228148956579, −6.30963939315570514625207605777, −5.54186781014840908201099865385, −4.43416078737319559841065809327, −3.84576798211450343629516951056, −2.97905561823466975471435164007, −1.95408864980247837229920486026, 1.06325820134013949483491311932, 2.70234917462913649689361468449, 3.60117112652933636439567058269, 4.46659912749071033888976749397, 5.02188476208891148704309135797, 5.98096505893238460380413214795, 6.68765116155070506204628021825, 7.68764651265525100590650019286, 8.652250866394158875610241704387, 9.342438841294776775116646485435

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