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Properties

Label 2-1323-63.25-c1-0-22
Degree $2$
Conductor $1323$
Sign $0.306 + 0.951i$
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  − 2.05·2-s + 2.21·4-s + (0.0731 − 0.126i)5-s − 0.446·8-s + (−0.150 + 0.260i)10-s + (0.832 + 1.44i)11-s + (−0.0999 − 0.173i)13-s − 3.51·16-s + (3.13 − 5.43i)17-s + (−3.45 − 5.99i)19-s + (0.162 − 0.280i)20-s + (−1.70 − 2.95i)22-s + (−3.09 + 5.35i)23-s + (2.48 + 4.31i)25-s + (0.205 + 0.355i)26-s + ⋯
L(s)  = 1  − 1.45·2-s + 1.10·4-s + (0.0327 − 0.0566i)5-s − 0.157·8-s + (−0.0474 + 0.0822i)10-s + (0.250 + 0.434i)11-s + (−0.0277 − 0.0480i)13-s − 0.879·16-s + (0.760 − 1.31i)17-s + (−0.793 − 1.37i)19-s + (0.0362 − 0.0627i)20-s + (−0.364 − 0.630i)22-s + (−0.644 + 1.11i)23-s + (0.497 + 0.862i)25-s + (0.0402 + 0.0697i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.306 + 0.951i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.306 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6302671443\)
\(L(\frac12)\) \(\approx\) \(0.6302671443\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.05T + 2T^{2} \)
5 \( 1 + (-0.0731 + 0.126i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.832 - 1.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.0999 + 0.173i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.13 + 5.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.45 + 5.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.09 - 5.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.46 + 4.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.51T + 31T^{2} \)
37 \( 1 + (3.50 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.15 - 2.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.940 - 1.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.81T + 47T^{2} \)
53 \( 1 + (-2.67 + 4.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.57T + 59T^{2} \)
61 \( 1 - 0.678T + 61T^{2} \)
67 \( 1 + 6.18T + 67T^{2} \)
71 \( 1 + 1.27T + 71T^{2} \)
73 \( 1 + (-0.778 + 1.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + (-3.75 + 6.50i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.53 - 7.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.98 + 6.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.434152463482522389735548514408, −8.859209418579498965447647189777, −7.892626030991502558573919731250, −7.28429277360283634612344687008, −6.57415488789324037322812911998, −5.30069861150181125753121517471, −4.37845012543263320094190154398, −2.94034253910136035704973129829, −1.79451483593929747994991413696, −0.50319860117707937861137646698, 1.11362482477422804298866672951, 2.18000532291193180183887116813, 3.59182598490873726723020540075, 4.64521548277797799546686123443, 6.07602049340738436375151577801, 6.56581696544068233236385889824, 7.73651248022800801628278970051, 8.398688375363215644562147785457, 8.726811883280255251818078010629, 9.911877336921743199732601976272

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