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Properties

Label 2-420-21.5-c1-0-0
Degree $2$
Conductor $420$
Sign $-0.999 + 0.0214i$
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  + (−0.317 + 1.70i)3-s + (−0.5 + 0.866i)5-s + (−1.73 + 1.99i)7-s + (−2.79 − 1.07i)9-s + (−3.38 + 1.95i)11-s − 6.06i·13-s + (−1.31 − 1.12i)15-s + (1.53 + 2.65i)17-s + (−2.94 − 1.70i)19-s + (−2.85 − 3.58i)21-s + (2.48 + 1.43i)23-s + (−0.499 − 0.866i)25-s + (2.72 − 4.42i)27-s + 7.97i·29-s + (−5.63 + 3.25i)31-s + ⋯
L(s)  = 1  + (−0.183 + 0.983i)3-s + (−0.223 + 0.387i)5-s + (−0.655 + 0.755i)7-s + (−0.932 − 0.359i)9-s + (−1.01 + 0.588i)11-s − 1.68i·13-s + (−0.339 − 0.290i)15-s + (0.371 + 0.643i)17-s + (−0.676 − 0.390i)19-s + (−0.622 − 0.782i)21-s + (0.518 + 0.299i)23-s + (−0.0999 − 0.173i)25-s + (0.524 − 0.851i)27-s + 1.48i·29-s + (−1.01 + 0.583i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.999 + 0.0214i$
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: \((2,\ 420,\ (\ :1/2),\ -0.999 + 0.0214i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00600227 - 0.559677i\)
\(L(\frac12)\) \(\approx\) \(0.00600227 - 0.559677i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.317 - 1.70i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (1.73 - 1.99i)T \)
good11 \( 1 + (3.38 - 1.95i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.06iT - 13T^{2} \)
17 \( 1 + (-1.53 - 2.65i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.94 + 1.70i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.48 - 1.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.97iT - 29T^{2} \)
31 \( 1 + (5.63 - 3.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0654 - 0.113i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 12.3T + 41T^{2} \)
43 \( 1 - 4.43T + 43T^{2} \)
47 \( 1 + (5.02 - 8.71i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.64 + 2.67i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.28 - 2.23i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.44 - 4.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.99 - 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.63iT - 71T^{2} \)
73 \( 1 + (6.72 - 3.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.22 - 2.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.63T + 83T^{2} \)
89 \( 1 + (-4.11 + 7.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.74iT - 97T^{2} \)
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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

−11.46127960278328514991379595974, −10.40727921228703227683857791630, −10.22513691557364121224163018461, −8.971881815474909897488293378201, −8.180002195131419426844509261775, −6.94309282491779803469760608239, −5.65674540636146084235780227270, −5.09898195152646040534756797008, −3.52812242662507345920565425287, −2.74732071099013851919386790538, 0.34527857530844870470742976559, 2.11979692804278955516303235664, 3.62974626208302739417572836085, 4.97367580413474176774566708239, 6.18997587007253843473720596107, 6.99769869302029371740029497376, 7.85414889773566106569620609240, 8.773170857822552153483061054624, 9.833652307361385973705387347223, 10.94597846021065628461767442417

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