Properties

Label 2-1620-180.167-c0-0-3
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: $\chi_{1620} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.548 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.086399401\)
\(L(\frac12)\) \(\approx\) \(2.086399401\)
\(L(1)\) 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

−10.04128762777582076569372110649, −8.743853717371837974354219215498, −7.83651076139642707586995853827, −7.13951472902241617781450168332, −6.47215399947650069250528863775, −5.55911920194567413784618488031, −4.95069704406478965128708288720, −3.59355093683939764285563496747, −3.05051686259952760547042442518, −1.98351477667380512182141178703, 1.43293737938939737523629963426, 2.38256177643502493159938384063, 3.67997943357391030786871549092, 4.60712475163060111899372708999, 5.05778694932925308541634052654, 6.17593097403503550775060776019, 6.74214661012656216437784750582, 7.82988219685914492322175405905, 8.717378325915981991698902459332, 9.656731080358730846113710143698

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