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Properties

Label 2-420-420.419-c0-0-3
Degree $2$
Conductor $420$
Sign $-1$
Analytic cond. $0.209607$
Root an. cond. $0.457828$
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·2-s − i·3-s − 4-s − 5-s − 6-s − i·7-s + i·8-s − 9-s + i·10-s + i·12-s − 14-s + i·15-s + 16-s + i·18-s + 20-s − 21-s + ⋯
L(s)  = 1  − i·2-s − i·3-s − 4-s − 5-s − 6-s − i·7-s + i·8-s − 9-s + i·10-s + i·12-s − 14-s + i·15-s + 16-s + i·18-s + 20-s − 21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(0.209607\)
Root analytic conductor: \(0.457828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5774576591\)
\(L(\frac12)\) \(\approx\) \(0.5774576591\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + iT \)
5 \( 1 + T \)
7 \( 1 + iT \)
good11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 2iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−11.02650155746776592515764124773, −10.47702528357781908294175473677, −9.080090020376538106466831649773, −8.171952976888622737909869378213, −7.48718131269717876772460119877, −6.37855064486552610819181081132, −4.78738509771929236619495438006, −3.80714794609498701627206570596, −2.57544418646657357988802409667, −0.829579921432508748114186712288, 3.20239717745019535536638769382, 4.20759500455270168954040228116, 5.22305837920699665425129723876, 6.01417330079669032641365035135, 7.39997322552566837785584800773, 8.234637166994617150640613667253, 9.074599896973942611869709502385, 9.701440309508268087520475044691, 10.98661769283825681538507651743, 11.79039900547583886584046335973

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