Properties

Label 2-1620-180.83-c0-0-3
Degree $2$
Conductor $1620$
Sign $0.746 + 0.665i$
Analytic cond. $0.808485$
Root an. cond. $0.899158$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.965 − 0.258i)5-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)10-s + (0.5 − 1.86i)13-s + (0.500 − 0.866i)16-s + (−1.22 − 1.22i)17-s + (0.707 − 0.707i)20-s + (0.866 − 0.499i)25-s + 1.93i·26-s + (−0.258 + 0.448i)29-s + (−0.258 + 0.965i)32-s + (1.49 + 0.866i)34-s + (−0.366 + 0.366i)37-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.965 − 0.258i)5-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)10-s + (0.5 − 1.86i)13-s + (0.500 − 0.866i)16-s + (−1.22 − 1.22i)17-s + (0.707 − 0.707i)20-s + (0.866 − 0.499i)25-s + 1.93i·26-s + (−0.258 + 0.448i)29-s + (−0.258 + 0.965i)32-s + (1.49 + 0.866i)34-s + (−0.366 + 0.366i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.746 + 0.665i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.746 + 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8355299361\)
\(L(\frac12)\) \(\approx\) \(0.8355299361\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (-0.965 + 0.258i)T \)
good7 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
41 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T^{2} \)
89 \( 1 - 1.93T + T^{2} \)
97 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.304740275664754218788024836435, −8.887187359021881249393629806634, −7.996327798648295772361188084775, −7.20528156483661560622247204300, −6.29833154706224157698784039949, −5.61771033439227744694634092171, −4.85124731266800994109331965306, −3.11437584173295641246290113908, −2.26238296096108994693738257683, −0.918785315892617106326793551235, 1.67719598762745788912517095488, 2.20683038634058513972699372644, 3.57106730456090437439555961658, 4.58268216500588272182625492197, 6.17953680907566618258237410787, 6.37354753162550699366786992545, 7.28147449679951386022492170156, 8.357403317411204033660568167858, 9.080676676698046781978875771712, 9.487068460630137538867667127125

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