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Properties

Label 2-420-140.139-c1-0-15
Degree $2$
Conductor $420$
Sign $0.895 + 0.444i$
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 + (2.00 − 2.44i)10-s + 2.86i·11-s + (0.452 + 1.94i)12-s + 4.36·13-s + (3.64 − 0.844i)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 + (0.635 − 0.772i)10-s + 0.864i·11-s + (0.130 + 0.562i)12-s + 1.21·13-s + (0.974 − 0.225i)14-s + (0.415 − 0.401i)15-s + (0.897 − 0.440i)16-s + 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.895 + 0.444i$
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: \((2,\ 420,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30820 - 0.306803i\)
\(L(\frac12)\) \(\approx\) \(1.30820 - 0.306803i\)
\(L(\frac{3}{2})\) 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

−11.18496427925769587157568696177, −10.25340959133962972866422897538, −9.528561569819347642495626340812, −8.531945351506652742644459560545, −7.63976710969836040238649542988, −6.19204025683895696506903231579, −5.49866781123433149567533088458, −3.86219439879898698249064839832, −2.56614788791229653232895780666, −1.69220356913218557272511276549, 1.03915235366114532973840732431, 3.66807254590836127155571714215, 4.51371085339752357884845404163, 5.77973980344658072349375856741, 6.21700843896019172181378953026, 7.81340360600459761143721249981, 8.399664526865478967084403201814, 9.380410554761177406099431244745, 10.15841374012918674188968880153, 10.90287664226336706122045301617

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