Properties

Label 2-420-140.139-c1-0-23
Degree $2$
Conductor $420$
Sign $0.608 + 0.793i$
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  + (0.161 − 1.40i)2-s + i·3-s + (−1.94 − 0.452i)4-s + (1.55 + 1.60i)5-s + (1.40 + 0.161i)6-s + (−0.299 − 2.62i)7-s + (−0.949 + 2.66i)8-s − 9-s + (2.50 − 1.92i)10-s − 2.86i·11-s + (0.452 − 1.94i)12-s + 4.36·13-s + (−3.74 − 0.00187i)14-s + (−1.60 + 1.55i)15-s + (3.59 + 1.76i)16-s + 5.54·17-s + ⋯
L(s)  = 1  + (0.113 − 0.993i)2-s + 0.577i·3-s + (−0.974 − 0.226i)4-s + (0.694 + 0.719i)5-s + (0.573 + 0.0657i)6-s + (−0.113 − 0.993i)7-s + (−0.335 + 0.941i)8-s − 0.333·9-s + (0.793 − 0.608i)10-s − 0.864i·11-s + (0.130 − 0.562i)12-s + 1.21·13-s + (−0.999 − 0.000501i)14-s + (−0.415 + 0.401i)15-s + (0.897 + 0.440i)16-s + 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.608 + 0.793i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37730 - 0.679901i\)
\(L(\frac12)\) \(\approx\) \(1.37730 - 0.679901i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.161 + 1.40i)T \)
3 \( 1 - iT \)
5 \( 1 + (-1.55 - 1.60i)T \)
7 \( 1 + (0.299 + 2.62i)T \)
good11 \( 1 + 2.86iT - 11T^{2} \)
13 \( 1 - 4.36T + 13T^{2} \)
17 \( 1 - 5.54T + 17T^{2} \)
19 \( 1 - 2.90T + 19T^{2} \)
23 \( 1 - 7.73T + 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 + 6.16T + 31T^{2} \)
37 \( 1 + 4.76iT - 37T^{2} \)
41 \( 1 - 1.97iT - 41T^{2} \)
43 \( 1 + 12.6T + 43T^{2} \)
47 \( 1 - 6.78iT - 47T^{2} \)
53 \( 1 + 7.68iT - 53T^{2} \)
59 \( 1 - 4.30T + 59T^{2} \)
61 \( 1 - 8.05iT - 61T^{2} \)
67 \( 1 + 7.75T + 67T^{2} \)
71 \( 1 - 0.551iT - 71T^{2} \)
73 \( 1 + 4.49T + 73T^{2} \)
79 \( 1 - 3.07iT - 79T^{2} \)
83 \( 1 + 7.45iT - 83T^{2} \)
89 \( 1 + 2.91iT - 89T^{2} \)
97 \( 1 - 9.73T + 97T^{2} \)
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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

−10.99539404726757575329637538200, −10.33695646710001648070164544191, −9.573430548002603603460381131476, −8.703713909367774981482422456876, −7.41661597615275886059774876194, −6.02106342056161128166118109326, −5.17924199240826450022854103847, −3.59826215813528407285650528000, −3.22092134897339638510458289316, −1.26270443744718096919854412898, 1.45552312840941331891721677217, 3.33860613623280075699902616180, 5.06849823016754008647077755962, 5.59539658555037144953567593950, 6.55289265566450212871987517347, 7.58202574362436750648215006024, 8.603712910317505812803273152685, 9.175907580153698269914561304546, 10.05130112920201579745744154238, 11.64782170268903737667344725350

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