Properties

Label 2-1620-180.47-c0-0-5
Degree $2$
Conductor $1620$
Sign $-0.619 + 0.784i$
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  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (0.707 + 0.707i)8-s + (−0.866 − 0.500i)10-s + (−1.86 − 0.5i)13-s + (0.500 − 0.866i)16-s + (1.22 − 1.22i)17-s + (−0.258 + 0.965i)20-s − 1.00i·25-s + 1.93i·26-s + (0.258 − 0.448i)29-s + (−0.965 − 0.258i)32-s + (−1.49 − 0.866i)34-s + (−0.366 − 0.366i)37-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (0.707 + 0.707i)8-s + (−0.866 − 0.500i)10-s + (−1.86 − 0.5i)13-s + (0.500 − 0.866i)16-s + (1.22 − 1.22i)17-s + (−0.258 + 0.965i)20-s − 1.00i·25-s + 1.93i·26-s + (0.258 − 0.448i)29-s + (−0.965 − 0.258i)32-s + (−1.49 − 0.866i)34-s + (−0.366 − 0.366i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.619 + 0.784i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ -0.619 + 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9142795292\)
\(L(\frac12)\) \(\approx\) \(0.9142795292\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
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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

−9.536258371323119232304004031426, −8.812083528672338268737140619345, −7.79410049784833577651000707549, −7.23549380074647655760523003882, −5.64007064082226334073760499375, −5.11827698508651025267893950164, −4.28413145366950377734966843195, −2.93127897965490121460443675320, −2.23196682838475065824164388948, −0.816665320780349375061193051129, 1.66423372131465339649807052837, 2.96442023412964206608957295311, 4.23533263402240901755350994897, 5.20945746278870259014187012133, 5.91326880660120794987454385937, 6.72976824225190660670785213105, 7.40752119965898652786528859510, 8.081758924217614587665030471800, 9.155159563184182508098859232370, 9.824631826796653909743267085934

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