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Properties

Label 2-420-21.5-c1-0-4
Degree $2$
Conductor $420$
Sign $0.862 - 0.505i$
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  + (1.71 + 0.268i)3-s + (−0.5 + 0.866i)5-s + (2.57 + 0.615i)7-s + (2.85 + 0.919i)9-s + (−1.80 + 1.04i)11-s − 0.245i·13-s + (−1.08 + 1.34i)15-s + (−0.471 − 0.816i)17-s + (−0.465 − 0.268i)19-s + (4.23 + 1.74i)21-s + (2.40 + 1.38i)23-s + (−0.499 − 0.866i)25-s + (4.63 + 2.34i)27-s + 0.267i·29-s + (0.981 − 0.566i)31-s + ⋯
L(s)  = 1  + (0.987 + 0.155i)3-s + (−0.223 + 0.387i)5-s + (0.972 + 0.232i)7-s + (0.951 + 0.306i)9-s + (−0.544 + 0.314i)11-s − 0.0681i·13-s + (−0.281 + 0.347i)15-s + (−0.114 − 0.198i)17-s + (−0.106 − 0.0616i)19-s + (0.924 + 0.380i)21-s + (0.500 + 0.288i)23-s + (−0.0999 − 0.173i)25-s + (0.892 + 0.450i)27-s + 0.0496i·29-s + (0.176 − 0.101i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.862 - 0.505i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.862 - 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90760 + 0.517670i\)
\(L(\frac12)\) \(\approx\) \(1.90760 + 0.517670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.71 - 0.268i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.57 - 0.615i)T \)
good11 \( 1 + (1.80 - 1.04i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.245iT - 13T^{2} \)
17 \( 1 + (0.471 + 0.816i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.465 + 0.268i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.40 - 1.38i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.267iT - 29T^{2} \)
31 \( 1 + (-0.981 + 0.566i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.08 + 5.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.38T + 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + (-6.23 + 10.7i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.8 - 6.26i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.25 + 10.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.96 - 2.86i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.78 - 4.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (11.3 - 6.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.17 + 5.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.06T + 83T^{2} \)
89 \( 1 + (0.463 - 0.803i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.01iT - 97T^{2} \)
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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

−11.14703116592453898178347446258, −10.36327971568879156540835922974, −9.383619872180099610960339429510, −8.433914967231817865258510801272, −7.74644688954867478651512856911, −6.89148976159441901704650490839, −5.30049858500172511589001789337, −4.32712983831205362712263160039, −3.05856699418363121587731629157, −1.90573626261409364763706256387, 1.44506140213492143628961936886, 2.87336925240382479385817478466, 4.18811312919111287669730180150, 5.09063314144309457975855108982, 6.61173908064849180990747137749, 7.77258390955147849956447507967, 8.252876071221978711587904571377, 9.103465767981115710182750547608, 10.17594862550916904364879673129, 11.08491982111414002756564409799

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