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Properties

Label 2-420-12.11-c1-0-41
Degree $2$
Conductor $420$
Sign $0.988 + 0.151i$
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  + (1.39 − 0.239i)2-s + (1.52 + 0.820i)3-s + (1.88 − 0.668i)4-s − i·5-s + (2.32 + 0.776i)6-s − i·7-s + (2.46 − 1.38i)8-s + (1.65 + 2.50i)9-s + (−0.239 − 1.39i)10-s − 3.82·11-s + (3.42 + 0.525i)12-s − 1.76·13-s + (−0.239 − 1.39i)14-s + (0.820 − 1.52i)15-s + (3.10 − 2.52i)16-s + 2.78i·17-s + ⋯
L(s)  = 1  + (0.985 − 0.169i)2-s + (0.880 + 0.473i)3-s + (0.942 − 0.334i)4-s − 0.447i·5-s + (0.948 + 0.317i)6-s − 0.377i·7-s + (0.872 − 0.489i)8-s + (0.551 + 0.834i)9-s + (−0.0758 − 0.440i)10-s − 1.15·11-s + (0.988 + 0.151i)12-s − 0.488·13-s + (−0.0641 − 0.372i)14-s + (0.211 − 0.393i)15-s + (0.776 − 0.630i)16-s + 0.676i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.988 + 0.151i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.988 + 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.08476 - 0.235326i\)
\(L(\frac12)\) \(\approx\) \(3.08476 - 0.235326i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.239i)T \)
3 \( 1 + (-1.52 - 0.820i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
good11 \( 1 + 3.82T + 11T^{2} \)
13 \( 1 + 1.76T + 13T^{2} \)
17 \( 1 - 2.78iT - 17T^{2} \)
19 \( 1 - 0.441iT - 19T^{2} \)
23 \( 1 + 6.19T + 23T^{2} \)
29 \( 1 + 4.99iT - 29T^{2} \)
31 \( 1 - 7.28iT - 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 - 2.13iT - 41T^{2} \)
43 \( 1 - 5.67iT - 43T^{2} \)
47 \( 1 + 7.38T + 47T^{2} \)
53 \( 1 + 10.6iT - 53T^{2} \)
59 \( 1 + 7.27T + 59T^{2} \)
61 \( 1 + 0.314T + 61T^{2} \)
67 \( 1 + 5.23iT - 67T^{2} \)
71 \( 1 - 3.60T + 71T^{2} \)
73 \( 1 + 1.17T + 73T^{2} \)
79 \( 1 + 13.0iT - 79T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 + 6.06iT - 89T^{2} \)
97 \( 1 + 1.71T + 97T^{2} \)
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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

−11.14476797795058120339558617234, −10.24719272110587348517481531845, −9.679379630456026545549829147818, −8.143601508592421648477766164650, −7.68625687687844566274361578225, −6.25666480696344237384517633611, −5.03467691533898174042762668946, −4.29249869091942666200384512780, −3.16039478468936421758427120515, −1.98613983509183172491760558118, 2.23930804465605129288846555708, 2.93408994422238335068511261766, 4.22143223525473985320579366541, 5.48887983977130631027560071455, 6.51603900288019240937426115454, 7.56639529081387614377327205737, 8.017111765769867655968656419950, 9.409028351636819248764150126496, 10.39599730436150662164828554871, 11.49172403278895746193442284329

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