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Properties

Label 2-420-105.2-c1-0-2
Degree $2$
Conductor $420$
Sign $0.0399 - 0.999i$
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  + (−0.396 − 1.68i)3-s + (−0.133 + 2.23i)5-s + (0.576 + 2.58i)7-s + (−2.68 + 1.33i)9-s + (−3.73 + 2.15i)11-s + (−2 + 2i)13-s + (3.81 − 0.658i)15-s + (−5.89 − 1.57i)17-s + (5.47 + 3.15i)19-s + (4.12 − 1.99i)21-s + (2.94 − 0.789i)23-s + (−4.96 − 0.598i)25-s + (3.31 + 4i)27-s − 0.683·29-s + (3 + 5.19i)31-s + ⋯
L(s)  = 1  + (−0.228 − 0.973i)3-s + (−0.0599 + 0.998i)5-s + (0.217 + 0.976i)7-s + (−0.895 + 0.445i)9-s + (−1.12 + 0.650i)11-s + (−0.554 + 0.554i)13-s + (0.985 − 0.169i)15-s + (−1.43 − 0.383i)17-s + (1.25 + 0.724i)19-s + (0.900 − 0.435i)21-s + (0.614 − 0.164i)23-s + (−0.992 − 0.119i)25-s + (0.638 + 0.769i)27-s − 0.126·29-s + (0.538 + 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0399 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0399 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.0399 - 0.999i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.0399 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.615753 + 0.591622i\)
\(L(\frac12)\) \(\approx\) \(0.615753 + 0.591622i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.396 + 1.68i)T \)
5 \( 1 + (0.133 - 2.23i)T \)
7 \( 1 + (-0.576 - 2.58i)T \)
good11 \( 1 + (3.73 - 2.15i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 - 2i)T - 13iT^{2} \)
17 \( 1 + (5.89 + 1.57i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.47 - 3.15i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.94 + 0.789i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 0.683T + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.46 + 1.46i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.63iT - 41T^{2} \)
43 \( 1 + (-5.15 + 5.15i)T - 43iT^{2} \)
47 \( 1 + (1.83 + 6.83i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.250 + 0.933i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.84 - 4.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.34 - 2.32i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.40 + 5.24i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.31iT - 71T^{2} \)
73 \( 1 + (9.06 + 2.42i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (10.6 + 6.15i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.15 - 7.15i)T + 83iT^{2} \)
89 \( 1 + (8.29 - 14.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.68 + 2.68i)T + 97iT^{2} \)
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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

−11.55610122735103072030976788697, −10.71804157131365008911813407026, −9.633771245051983835508334868493, −8.518060970742877285497157140620, −7.48839359522566006953597443980, −6.91214549045996390382284853221, −5.82829549779792449634942546248, −4.85733225547499695304105651509, −2.85294056966667305655501715685, −2.12468489913582060542141138428, 0.54057259988327982743128590890, 2.91902778529854106208448631626, 4.31931341749693028479848785304, 4.94929592768629407660431269118, 5.91035850161260926981981620784, 7.46197440685939907717249946934, 8.318267213251444302250284502837, 9.282862194584277679105841580274, 10.06469676898167721511606874109, 11.02332529139667081579406558920

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