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Properties

Label 2-1620-180.167-c0-0-2
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: $\chi_{1620} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.746 - 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.822443214\)
\(L(\frac12)\) \(\approx\) \(1.822443214\)
\(L(1)\) 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

−9.577299949292323914621764842667, −8.721379209449021083528438756253, −7.914571285032784022255580883283, −7.08929890066321207634765751633, −6.60276728275919689309488735449, −5.37530661828103852456376742454, −4.69084544375554521517513565773, −3.84694304241073302815922391237, −3.10632177415839527912161308970, −1.66228475433955525174610720029, 1.25719470074902281596469105692, 2.95608420041304829326691641101, 3.43723002873263752008228732677, 4.35164164595997464062830048294, 5.40200484753564014770791763470, 6.04519781778741095813716057039, 7.01055489132790251654518729923, 7.987136880002714804208652749874, 8.267529327029634857449384159745, 9.886570933456060098006701164930

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