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Properties

Label 2-1620-180.83-c0-0-0
Degree $2$
Conductor $1620$
Sign $-0.548 - 0.835i$
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 + (−0.258 + 0.965i)5-s + (−0.707 + 0.707i)8-s − i·10-s + (−0.366 + 1.36i)13-s + (0.500 − 0.866i)16-s + (0.258 + 0.965i)20-s + (−0.866 − 0.499i)25-s − 1.41i·26-s + (−0.707 + 1.22i)29-s + (−0.258 + 0.965i)32-s + (−1 + i)37-s + (−0.5 − 0.866i)40-s + (1.22 − 0.707i)41-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.258 + 0.965i)5-s + (−0.707 + 0.707i)8-s − i·10-s + (−0.366 + 1.36i)13-s + (0.500 − 0.866i)16-s + (0.258 + 0.965i)20-s + (−0.866 − 0.499i)25-s − 1.41i·26-s + (−0.707 + 1.22i)29-s + (−0.258 + 0.965i)32-s + (−1 + i)37-s + (−0.5 − 0.866i)40-s + (1.22 − 0.707i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.548 - 0.835i$
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: \((2,\ 1620,\ (\ :0),\ -0.548 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5382975431\)
\(L(\frac12)\) \(\approx\) \(0.5382975431\)
\(L(1)\) 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 \( 1 + (0.258 - 0.965i)T \)
good7 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
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 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
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

−9.731340987448029854701496864949, −9.136593175126086093083957831853, −8.274001080976522607982228310549, −7.36206678453126129196598537334, −6.88392466371099360546275962029, −6.18280521804576523509762220351, −5.08446267468287852723430880817, −3.81187737912489637959534621995, −2.71143388325920194190630572401, −1.70232108226614262334799508408, 0.56277024507619513247984376814, 1.92965298235769861204746495724, 3.11911721403236545769765482694, 4.14110564871930074621456468835, 5.31667075219334158662812883099, 6.08591603271269390542203648919, 7.31247195029840901605642585698, 7.903066882469458247115470395351, 8.493429447916081865194537877393, 9.369707911908180682863424241802

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