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Properties

Label 2-420-60.59-c1-0-21
Degree $2$
Conductor $420$
Sign $-0.0695 + 0.997i$
Analytic cond. $3.35371$
Root an. cond. $1.83131$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.979 − 1.02i)2-s + (−1.71 − 0.218i)3-s + (−0.0819 + 1.99i)4-s + (−2.22 + 0.218i)5-s + (1.45 + 1.96i)6-s + 7-s + (2.11 − 1.87i)8-s + (2.90 + 0.750i)9-s + (2.40 + 2.05i)10-s − 2.59·11-s + (0.577 − 3.41i)12-s + 5.21i·13-s + (−0.979 − 1.02i)14-s + (3.87 + 0.110i)15-s + (−3.98 − 0.327i)16-s + 3.91·17-s + ⋯
L(s)  = 1  + (−0.692 − 0.721i)2-s + (−0.992 − 0.126i)3-s + (−0.0409 + 0.999i)4-s + (−0.995 + 0.0976i)5-s + (0.595 + 0.803i)6-s + 0.377·7-s + (0.749 − 0.662i)8-s + (0.968 + 0.250i)9-s + (0.759 + 0.650i)10-s − 0.781·11-s + (0.166 − 0.986i)12-s + 1.44i·13-s + (−0.261 − 0.272i)14-s + (0.999 + 0.0286i)15-s + (−0.996 − 0.0819i)16-s + 0.950·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0695 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0695 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.0695 + 0.997i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.0695 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332309 - 0.356289i\)
\(L(\frac12)\) \(\approx\) \(0.332309 - 0.356289i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.979 + 1.02i)T \)
3 \( 1 + (1.71 + 0.218i)T \)
5 \( 1 + (2.22 - 0.218i)T \)
7 \( 1 - T \)
good11 \( 1 + 2.59T + 11T^{2} \)
13 \( 1 - 5.21iT - 13T^{2} \)
17 \( 1 - 3.91T + 17T^{2} \)
19 \( 1 + 5.21iT - 19T^{2} \)
23 \( 1 + 7.44iT - 23T^{2} \)
29 \( 1 + 8.44iT - 29T^{2} \)
31 \( 1 + 1.42iT - 31T^{2} \)
37 \( 1 + 7.99iT - 37T^{2} \)
41 \( 1 - 7.49iT - 41T^{2} \)
43 \( 1 - 3.28T + 43T^{2} \)
47 \( 1 - 4.36iT - 47T^{2} \)
53 \( 1 + 0.310T + 53T^{2} \)
59 \( 1 - 2.17T + 59T^{2} \)
61 \( 1 - 6.70T + 61T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 - 5.88T + 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 + 7.09iT - 79T^{2} \)
83 \( 1 - 0.802iT - 83T^{2} \)
89 \( 1 + 6.98iT - 89T^{2} \)
97 \( 1 + 6.80iT - 97T^{2} \)
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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

−11.17061810225289801187377570822, −10.31528932434863741403868398048, −9.291545151229384260246595330566, −8.139066659773861432910890692939, −7.42075958428121937000408557103, −6.49732892763253092129283073568, −4.80596796247445629241940931173, −4.08490222112926353840953389430, −2.38758534220271275354055975338, −0.56782448380520726439078192272, 1.09024680608640426700690187883, 3.63063418445302744094841976884, 5.29770474077448760936383832148, 5.42687045717768663619443725524, 6.98475491226966629481852199955, 7.76416542762094707062573114398, 8.341712988030124702625936815325, 9.826398109677484537583422631277, 10.48479653877268899491746809566, 11.18686129924864932658744544605

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