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Properties

Label 2-420-28.27-c1-0-6
Degree $2$
Conductor $420$
Sign $0.712 - 0.701i$
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 − 0.238i)2-s − 3-s + (1.88 + 0.664i)4-s + i·5-s + (1.39 + 0.238i)6-s + (2.37 + 1.16i)7-s + (−2.47 − 1.37i)8-s + 9-s + (0.238 − 1.39i)10-s − 4.86i·11-s + (−1.88 − 0.664i)12-s + 3.63i·13-s + (−3.03 − 2.18i)14-s − i·15-s + (3.11 + 2.50i)16-s + 4.47i·17-s + ⋯
L(s)  = 1  + (−0.985 − 0.168i)2-s − 0.577·3-s + (0.943 + 0.332i)4-s + 0.447i·5-s + (0.569 + 0.0973i)6-s + (0.898 + 0.439i)7-s + (−0.873 − 0.486i)8-s + 0.333·9-s + (0.0754 − 0.440i)10-s − 1.46i·11-s + (−0.544 − 0.191i)12-s + 1.00i·13-s + (−0.811 − 0.584i)14-s − 0.258i·15-s + (0.779 + 0.627i)16-s + 1.08i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.712 - 0.701i$
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: \((2,\ 420,\ (\ :1/2),\ 0.712 - 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.726519 + 0.297471i\)
\(L(\frac12)\) \(\approx\) \(0.726519 + 0.297471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.238i)T \)
3 \( 1 + T \)
5 \( 1 - iT \)
7 \( 1 + (-2.37 - 1.16i)T \)
good11 \( 1 + 4.86iT - 11T^{2} \)
13 \( 1 - 3.63iT - 13T^{2} \)
17 \( 1 - 4.47iT - 17T^{2} \)
19 \( 1 + 2.70T + 19T^{2} \)
23 \( 1 - 1.68iT - 23T^{2} \)
29 \( 1 - 8.31T + 29T^{2} \)
31 \( 1 - 5.47T + 31T^{2} \)
37 \( 1 - 7.07T + 37T^{2} \)
41 \( 1 - 11.5iT - 41T^{2} \)
43 \( 1 - 7.86iT - 43T^{2} \)
47 \( 1 - 4.75T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + 2.97T + 59T^{2} \)
61 \( 1 + 1.18iT - 61T^{2} \)
67 \( 1 + 13.1iT - 67T^{2} \)
71 \( 1 - 14.8iT - 71T^{2} \)
73 \( 1 - 1.40iT - 73T^{2} \)
79 \( 1 + 1.01iT - 79T^{2} \)
83 \( 1 - 8.22T + 83T^{2} \)
89 \( 1 + 9.91iT - 89T^{2} \)
97 \( 1 + 13.0iT - 97T^{2} \)
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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

−11.25322236165551790000360352438, −10.59305214291680654386816495201, −9.534996808401703939519979703843, −8.439998688044666034847251968678, −7.963564300997805775467527608078, −6.46509329567252742365140339631, −6.06694660434010031963265759178, −4.42320104430651061040028890885, −2.86126561138606345524060249603, −1.38249597964178905633548292033, 0.846604608330649069106960735003, 2.34143240328813975621932900583, 4.49661942604033654962550179408, 5.32428923329039678664404537953, 6.63953333749934121098205545814, 7.50037995010502966786654460960, 8.237705952805371070649493448111, 9.337365526821412237200971547434, 10.29067435651677557420966211967, 10.76718277225744476405620701910

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