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Properties

Label 2-420-28.27-c1-0-9
Degree $2$
Conductor $420$
Sign $-0.355 - 0.934i$
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  + (1.10 + 0.887i)2-s − 3-s + (0.426 + 1.95i)4-s − i·5-s + (−1.10 − 0.887i)6-s + (0.391 + 2.61i)7-s + (−1.26 + 2.53i)8-s + 9-s + (0.887 − 1.10i)10-s − 0.770i·11-s + (−0.426 − 1.95i)12-s + 5.60i·13-s + (−1.88 + 3.22i)14-s + i·15-s + (−3.63 + 1.66i)16-s + 0.503i·17-s + ⋯
L(s)  = 1  + (0.778 + 0.627i)2-s − 0.577·3-s + (0.213 + 0.977i)4-s − 0.447i·5-s + (−0.449 − 0.362i)6-s + (0.148 + 0.988i)7-s + (−0.446 + 0.894i)8-s + 0.333·9-s + (0.280 − 0.348i)10-s − 0.232i·11-s + (−0.123 − 0.564i)12-s + 1.55i·13-s + (−0.504 + 0.863i)14-s + 0.258i·15-s + (−0.909 + 0.416i)16-s + 0.122i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.355 - 0.934i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.355 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.945902 + 1.37191i\)
\(L(\frac12)\) \(\approx\) \(0.945902 + 1.37191i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.10 - 0.887i)T \)
3 \( 1 + T \)
5 \( 1 + iT \)
7 \( 1 + (-0.391 - 2.61i)T \)
good11 \( 1 + 0.770iT - 11T^{2} \)
13 \( 1 - 5.60iT - 13T^{2} \)
17 \( 1 - 0.503iT - 17T^{2} \)
19 \( 1 - 1.63T + 19T^{2} \)
23 \( 1 - 1.42iT - 23T^{2} \)
29 \( 1 - 5.03T + 29T^{2} \)
31 \( 1 + 8.23T + 31T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 + 5.07iT - 41T^{2} \)
43 \( 1 + 9.06iT - 43T^{2} \)
47 \( 1 - 4.64T + 47T^{2} \)
53 \( 1 - 0.455T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 3.32iT - 61T^{2} \)
67 \( 1 + 8.70iT - 67T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 + 2.29iT - 73T^{2} \)
79 \( 1 - 2.56iT - 79T^{2} \)
83 \( 1 - 13.9T + 83T^{2} \)
89 \( 1 + 2.98iT - 89T^{2} \)
97 \( 1 - 15.5iT - 97T^{2} \)
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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

−11.82832856392240228618222540479, −10.95535731002743503716424081087, −9.352606780867815670877988190901, −8.738473363415336831489470768062, −7.57744630039923829306029382794, −6.55033631849058626461599262131, −5.69983635600507597296465110860, −4.89195475064939283614489125547, −3.83452191219643485917080323817, −2.15348210069654411925720733699, 0.947990852725444972876303665244, 2.79388187550449142987065278866, 3.95316979528871867715805341903, 5.00241605493045378039210385021, 5.96960523846659849898185971040, 6.97604109661614274463337848207, 7.910828756599955015347517018453, 9.648866505405511549792014761466, 10.33227251700312640464458401077, 10.96493187729230058207145549010

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