Properties

Label 2-1620-5.3-c0-0-0
Degree $2$
Conductor $1620$
Sign $-0.850 - 0.525i$
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  + i·5-s + (−1 + i)7-s − 11-s + (−1 + i)17-s i·19-s + (−1 − i)23-s − 25-s + i·29-s + 31-s + (−1 − i)35-s − 41-s i·49-s + (1 + i)53-s i·55-s + i·59-s + ⋯
L(s)  = 1  + i·5-s + (−1 + i)7-s − 11-s + (−1 + i)17-s i·19-s + (−1 − i)23-s − 25-s + i·29-s + 31-s + (−1 − i)35-s − 41-s i·49-s + (1 + i)53-s i·55-s + i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ -0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5558104945\)
\(L(\frac12)\) \(\approx\) \(0.5558104945\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 + (1 - i)T - iT^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (1 - i)T - iT^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + iT - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−10.19737043187140351711801789303, −9.009293671166468499404791861215, −8.480971055099012828025218778166, −7.43077438352719075513547102893, −6.50752955969908720603522920448, −6.13953809572977233238603293969, −5.07487114245196319272723291505, −3.89968880573234909674772514469, −2.77690535333401520316094638848, −2.34158363970689025362885140343, 0.39798184820396681645332112451, 2.02801588623418918675538848601, 3.36450347605110785160769639157, 4.22266232440364836589613707871, 5.08060148370656358236704242125, 5.99615877641375413605742674760, 6.87441862660598577907019260160, 7.80050111670424135503191945327, 8.326325336335636022341127485430, 9.528469047834042388493293917643

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