Properties

Label 2-1782-9.4-c1-0-15
Degree $2$
Conductor $1782$
Sign $0.766 - 0.642i$
Analytic cond. $14.2293$
Root an. cond. $3.77217$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1 + 1.73i)5-s + (2 + 3.46i)7-s + 0.999·8-s + 1.99·10-s + (0.5 + 0.866i)11-s + (3 − 5.19i)13-s + (1.99 − 3.46i)14-s + (−0.5 − 0.866i)16-s + 2·17-s + 4·19-s + (−0.999 − 1.73i)20-s + (0.499 − 0.866i)22-s + (−2 + 3.46i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.447 + 0.774i)5-s + (0.755 + 1.30i)7-s + 0.353·8-s + 0.632·10-s + (0.150 + 0.261i)11-s + (0.832 − 1.44i)13-s + (0.534 − 0.925i)14-s + (−0.125 − 0.216i)16-s + 0.485·17-s + 0.917·19-s + (−0.223 − 0.387i)20-s + (0.106 − 0.184i)22-s + (−0.417 + 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1782\)    =    \(2 \cdot 3^{4} \cdot 11\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(14.2293\)
Root analytic conductor: \(3.77217\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1782} (595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1782,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.469590040\)
\(L(\frac12)\) \(\approx\) \(1.469590040\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
11 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.343205206338520900407985765824, −8.662491175949813007331539266276, −7.75383918100423541242416759766, −7.45892180871969647440146292828, −5.87026927332905148122852790369, −5.52683411646643825956499354129, −4.17729392851415610741626357656, −3.19444863872429356283236453158, −2.51016141691154590015405160160, −1.21137917161860257271172838558, 0.76445687512027243534363527773, 1.61974026695972400219071448857, 3.61561688605682570263444616910, 4.37223675632134724812614038870, 4.98084963502344651655389422168, 6.14359628000179431426173367395, 6.96137039359423436323101648569, 7.70165859237757295608375076862, 8.325251848935204919982984448834, 9.016091503447965786795765476800

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