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Properties

Label 2-1620-180.23-c0-0-3
Degree $2$
Conductor $1620$
Sign $-0.116 + 0.993i$
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 + (0.965 − 0.258i)5-s + (−0.707 + 0.707i)8-s − i·10-s + (1.36 − 0.366i)13-s + (0.500 + 0.866i)16-s + (−0.965 − 0.258i)20-s + (0.866 − 0.499i)25-s − 1.41i·26-s + (−0.707 − 1.22i)29-s + (0.965 − 0.258i)32-s + (−1 + i)37-s + (−0.5 + 0.866i)40-s + (−1.22 − 0.707i)41-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.965 − 0.258i)5-s + (−0.707 + 0.707i)8-s − i·10-s + (1.36 − 0.366i)13-s + (0.500 + 0.866i)16-s + (−0.965 − 0.258i)20-s + (0.866 − 0.499i)25-s − 1.41i·26-s + (−0.707 − 1.22i)29-s + (0.965 − 0.258i)32-s + (−1 + i)37-s + (−0.5 + 0.866i)40-s + (−1.22 − 0.707i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.116 + 0.993i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ -0.116 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.396021664\)
\(L(\frac12)\) \(\approx\) \(1.396021664\)
\(L(1)\) 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 \( 1 + (-0.965 + 0.258i)T \)
good7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.539220323972565347461712893859, −8.748551381579248897309712207305, −8.179252552574977565855141722177, −6.71675689901478247087168521493, −5.84489444316218614708802774368, −5.29255248698504494862476576280, −4.20054156482736399358291093034, −3.30024750514592060519041955445, −2.19970194892259404969061869648, −1.20796540875959255527920609997, 1.61856792214718953797928934948, 3.14060269571621754647289784448, 3.99234044952871783814205997437, 5.18951508748944023344378011625, 5.75362494449491974966513012783, 6.61533531209493834401351844932, 7.11365466041574077000080790625, 8.283057081664959249909226251833, 8.899725227233384053321336998874, 9.534794356523578891318366721253

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