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Properties

Label 2-420-420.419-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8561427586\)
\(L(\frac12)\) \(\approx\) \(0.8561427586\)
\(L(1)\) 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

−11.27774298483463422992297037087, −10.36318347351254437845862606871, −9.849601595597083540294523101414, −8.882506539365410725055741343258, −8.463863264496142831702407611184, −6.32033675389585685212976300560, −5.35941939797351213158440148615, −4.58372416401106402086980854062, −3.12699937113720365109990098080, −2.18555724450550750940052168903, 1.47449520356170165005542554715, 3.43455868965670459222249926831, 5.04149430345979851238375356586, 5.92501627286758805732609489839, 6.83816375826502816996740666978, 7.46992445989136278910409620516, 8.453314013034598148448683264804, 9.460954766192921269399508101495, 10.27690357648559879869450191358, 11.49167277879116772856096559277

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