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Properties

Label 2-6e3-72.13-c1-0-7
Degree $2$
Conductor $216$
Sign $0.422 + 0.906i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
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  + (−0.366 + 1.36i)2-s + (−1.73 − i)4-s + (−1.73 − i)5-s + (−2 − 3.46i)7-s + (2 − 1.99i)8-s + (2 − 1.99i)10-s + (2.59 − 1.5i)11-s + (−1.73 − i)13-s + (5.46 − 1.46i)14-s + (1.99 + 3.46i)16-s − 5·17-s + i·19-s + (1.99 + 3.46i)20-s + (1.09 + 4.09i)22-s + (1 − 1.73i)23-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.5i)4-s + (−0.774 − 0.447i)5-s + (−0.755 − 1.30i)7-s + (0.707 − 0.707i)8-s + (0.632 − 0.632i)10-s + (0.783 − 0.452i)11-s + (−0.480 − 0.277i)13-s + (1.46 − 0.391i)14-s + (0.499 + 0.866i)16-s − 1.21·17-s + 0.229i·19-s + (0.447 + 0.774i)20-s + (0.234 + 0.873i)22-s + (0.208 − 0.361i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.422 + 0.906i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.422 + 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.479225 - 0.305300i\)
\(L(\frac12)\) \(\approx\) \(0.479225 - 0.305300i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
3 \( 1 \)
good5 \( 1 + (1.73 + i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.73 + i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.52 + 5.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.3 - 6i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.46 + 2i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.32874166244514859434980186458, −11.00377825082115854740307636663, −10.01061177965729358495850629654, −9.021931076140402793235797727510, −8.027820702389442688812467601575, −7.07858802468233348800829322538, −6.25573032325991831145670480609, −4.60110141709492036974659436557, −3.80135711138864567527566904674, −0.52493485583485650390044569946, 2.29880230993597420210544542444, 3.49889497051532979730464336803, 4.77226171212387565280923462842, 6.42303801021836928459479527874, 7.63991990852734577553267530836, 9.032662994130590356893921312244, 9.359582965349885238221256654758, 10.73714426423818152221574443922, 11.64789701447128026064748700419, 12.22376269834268561879408511601

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