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Properties

Label 2-420-21.5-c1-0-8
Degree $2$
Conductor $420$
Sign $-0.597 + 0.802i$
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  + (−1.16 + 1.28i)3-s + (0.5 − 0.866i)5-s + (−1.80 − 1.93i)7-s + (−0.284 − 2.98i)9-s + (−4.05 + 2.34i)11-s + 2.18i·13-s + (0.527 + 1.64i)15-s + (−3.74 − 6.49i)17-s + (−0.638 − 0.368i)19-s + (4.58 − 0.0477i)21-s + (−6.99 − 4.03i)23-s + (−0.499 − 0.866i)25-s + (4.15 + 3.11i)27-s − 1.15i·29-s + (8.95 − 5.16i)31-s + ⋯
L(s)  = 1  + (−0.672 + 0.739i)3-s + (0.223 − 0.387i)5-s + (−0.680 − 0.732i)7-s + (−0.0947 − 0.995i)9-s + (−1.22 + 0.706i)11-s + 0.607i·13-s + (0.136 + 0.425i)15-s + (−0.909 − 1.57i)17-s + (−0.146 − 0.0845i)19-s + (0.999 − 0.0104i)21-s + (−1.45 − 0.842i)23-s + (−0.0999 − 0.173i)25-s + (0.800 + 0.599i)27-s − 0.214i·29-s + (1.60 − 0.928i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.597 + 0.802i$
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: \((2,\ 420,\ (\ :1/2),\ -0.597 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153316 - 0.305270i\)
\(L(\frac12)\) \(\approx\) \(0.153316 - 0.305270i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.16 - 1.28i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.80 + 1.93i)T \)
good11 \( 1 + (4.05 - 2.34i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.18iT - 13T^{2} \)
17 \( 1 + (3.74 + 6.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.638 + 0.368i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.99 + 4.03i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.15iT - 29T^{2} \)
31 \( 1 + (-8.95 + 5.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.30 + 3.99i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.43T + 41T^{2} \)
43 \( 1 + 9.24T + 43T^{2} \)
47 \( 1 + (4.34 - 7.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.03 - 4.06i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.48 + 6.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.13 - 2.96i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.691 - 1.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.26iT - 71T^{2} \)
73 \( 1 + (0.211 - 0.122i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.79 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.4T + 83T^{2} \)
89 \( 1 + (0.658 - 1.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.84iT - 97T^{2} \)
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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

−10.78988571249027117758398483560, −9.818387607805815210224356512722, −9.566529634553055482734262392313, −8.149933511276374108641021236840, −6.94491732626372120225683391021, −6.11506892818378893293481840544, −4.80945805453920576456138886182, −4.27658120876714361262025799638, −2.61687245340190922518673525995, −0.22000748975483652772892720898, 2.03923454789571647572753713555, 3.24564123595883957356765862671, 5.08302350631228834782717362919, 6.05629862816466641196557661344, 6.49473510690178106624582121253, 7.952658448165951414472091657345, 8.462740533034131030324990637837, 10.07106419456353228413178293126, 10.55387053314779908292887087397, 11.57881176935402047652489117853

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