Properties

Label 2-420-60.59-c1-0-42
Degree $2$
Conductor $420$
Sign $-0.0721 + 0.997i$
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  + (−1.40 − 0.114i)2-s + (−1.47 + 0.909i)3-s + (1.97 + 0.321i)4-s + (1 − 2i)5-s + (2.18 − 1.11i)6-s + 7-s + (−2.74 − 0.678i)8-s + (1.34 − 2.68i)9-s + (−1.63 + 2.70i)10-s − 2.94·11-s + (−3.20 + 1.32i)12-s − 1.36i·13-s + (−1.40 − 0.114i)14-s + (0.345 + 3.85i)15-s + (3.79 + 1.26i)16-s − 2.69·17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0806i)2-s + (−0.851 + 0.525i)3-s + (0.986 + 0.160i)4-s + (0.447 − 0.894i)5-s + (0.890 − 0.454i)6-s + 0.377·7-s + (−0.970 − 0.239i)8-s + (0.448 − 0.893i)9-s + (−0.517 + 0.855i)10-s − 0.888·11-s + (−0.924 + 0.381i)12-s − 0.378i·13-s + (−0.376 − 0.0304i)14-s + (0.0891 + 0.996i)15-s + (0.948 + 0.317i)16-s − 0.652·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0721 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0721 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.0721 + 0.997i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.0721 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360937 - 0.387995i\)
\(L(\frac12)\) \(\approx\) \(0.360937 - 0.387995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.114i)T \)
3 \( 1 + (1.47 - 0.909i)T \)
5 \( 1 + (-1 + 2i)T \)
7 \( 1 - T \)
good11 \( 1 + 2.94T + 11T^{2} \)
13 \( 1 + 1.36iT - 13T^{2} \)
17 \( 1 + 2.69T + 17T^{2} \)
19 \( 1 - 3.54iT - 19T^{2} \)
23 \( 1 + 7.18iT - 23T^{2} \)
29 \( 1 - 1.36iT - 29T^{2} \)
31 \( 1 + 8.09iT - 31T^{2} \)
37 \( 1 + 9.89iT - 37T^{2} \)
41 \( 1 + 5.89iT - 41T^{2} \)
43 \( 1 + 2.25T + 43T^{2} \)
47 \( 1 + 1.81iT - 47T^{2} \)
53 \( 1 - 13.7T + 53T^{2} \)
59 \( 1 - 6.80T + 59T^{2} \)
61 \( 1 + 6.55T + 61T^{2} \)
67 \( 1 + 8.46T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + 16.1iT - 79T^{2} \)
83 \( 1 - 5.83iT - 83T^{2} \)
89 \( 1 - 4.83iT - 89T^{2} \)
97 \( 1 - 3.70iT - 97T^{2} \)
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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

−10.60219009867320951895050422354, −10.26263411501852975571021467835, −9.175401867935342207652815213508, −8.455763810569520325586113589761, −7.39235685158977588348194001373, −6.09603221673408156285180657106, −5.38415341282425124349537645200, −4.14272816173143493316226540712, −2.21377286195363116400680136314, −0.51178340982906139357597110462, 1.62336414812565982789499454679, 2.83185997903388869420924430454, 5.03019503273708777749767764422, 6.04606231569944149998662540614, 6.93916261513559362323361500961, 7.52822686979080204281765577581, 8.642361291270927182542916153803, 9.875593423251038683379839967755, 10.53384377284008460966311283721, 11.32070708239260066305121472108

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