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Properties

Label 2-1620-180.23-c0-0-2
Degree $2$
Conductor $1620$
Sign $0.929 + 0.370i$
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 + (−0.258 + 0.965i)5-s + (−0.707 + 0.707i)8-s + (0.866 + 0.499i)10-s + (0.5 − 0.133i)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 − 0.517i·26-s + (0.965 + 1.67i)29-s + (0.965 − 0.258i)32-s + (1.49 − 0.866i)34-s + (1.36 − 1.36i)37-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.258 + 0.965i)5-s + (−0.707 + 0.707i)8-s + (0.866 + 0.499i)10-s + (0.5 − 0.133i)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 − 0.517i·26-s + (0.965 + 1.67i)29-s + (0.965 − 0.258i)32-s + (1.49 − 0.866i)34-s + (1.36 − 1.36i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.929 + 0.370i$
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: \((2,\ 1620,\ (\ :0),\ 0.929 + 0.370i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.117058227\)
\(L(\frac12)\) \(\approx\) \(1.117058227\)
\(L(1)\) 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 \( 1 + (0.258 - 0.965i)T \)
good7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
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 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T^{2} \)
89 \( 1 + 0.517T + T^{2} \)
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

−9.827662670983312077852286236267, −8.824624432353315491204095580461, −8.114004261657007278495986653231, −7.14814725575536269139480565069, −6.10038949561449461175693271433, −5.45275748155278883619487732614, −4.17208502997471564729117333241, −3.49430614829849182051417299783, −2.67570478822736688983299329055, −1.39470823985653510512484914424, 0.965030486179191402372036170762, 2.93847405574644109625604090064, 4.03421985853609216437593257469, 4.79197290336693778090174336727, 5.52272174931461232948648721057, 6.35429479717932728549719145442, 7.29234239113308497524519531286, 8.159236272876095489695784393761, 8.469044306586829606087028680262, 9.671516190651630286158575300203

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