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Properties

Label 2-420-140.139-c1-0-19
Degree $2$
Conductor $420$
Sign $0.0738 - 0.997i$
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  + (1.08 + 0.903i)2-s − i·3-s + (0.366 + 1.96i)4-s + (−0.660 + 2.13i)5-s + (0.903 − 1.08i)6-s + (2.64 − 0.110i)7-s + (−1.37 + 2.46i)8-s − 9-s + (−2.64 + 1.72i)10-s + 2.42i·11-s + (1.96 − 0.366i)12-s − 5.02·13-s + (2.97 + 2.26i)14-s + (2.13 + 0.660i)15-s + (−3.73 + 1.43i)16-s + 6.95·17-s + ⋯
L(s)  = 1  + (0.769 + 0.639i)2-s − 0.577i·3-s + (0.183 + 0.983i)4-s + (−0.295 + 0.955i)5-s + (0.368 − 0.444i)6-s + (0.999 − 0.0416i)7-s + (−0.487 + 0.873i)8-s − 0.333·9-s + (−0.837 + 0.546i)10-s + 0.732i·11-s + (0.567 − 0.105i)12-s − 1.39·13-s + (0.795 + 0.606i)14-s + (0.551 + 0.170i)15-s + (−0.932 + 0.359i)16-s + 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0738 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0738 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.0738 - 0.997i$
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: \((2,\ 420,\ (\ :1/2),\ 0.0738 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50377 + 1.39650i\)
\(L(\frac12)\) \(\approx\) \(1.50377 + 1.39650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.08 - 0.903i)T \)
3 \( 1 + iT \)
5 \( 1 + (0.660 - 2.13i)T \)
7 \( 1 + (-2.64 + 0.110i)T \)
good11 \( 1 - 2.42iT - 11T^{2} \)
13 \( 1 + 5.02T + 13T^{2} \)
17 \( 1 - 6.95T + 17T^{2} \)
19 \( 1 - 1.26T + 19T^{2} \)
23 \( 1 - 6.81T + 23T^{2} \)
29 \( 1 + 4.56T + 29T^{2} \)
31 \( 1 - 0.903T + 31T^{2} \)
37 \( 1 + 8.02iT - 37T^{2} \)
41 \( 1 - 3.33iT - 41T^{2} \)
43 \( 1 + 2.06T + 43T^{2} \)
47 \( 1 + 3.31iT - 47T^{2} \)
53 \( 1 + 4.77iT - 53T^{2} \)
59 \( 1 - 8.38T + 59T^{2} \)
61 \( 1 + 4.38iT - 61T^{2} \)
67 \( 1 - 8.25T + 67T^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + 8.69iT - 79T^{2} \)
83 \( 1 + 9.72iT - 83T^{2} \)
89 \( 1 - 13.6iT - 89T^{2} \)
97 \( 1 + 1.23T + 97T^{2} \)
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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

−11.74201926837198752113999982512, −10.78635788487893213729687484402, −9.568669240165225222536966568596, −8.121094802251547800400440197753, −7.38552381614247799000269236948, −7.06151073249857031327654112173, −5.62600339066760890816056069045, −4.80668969974734153297344029560, −3.40689279428308569851632014609, −2.22529312411290457380034903642, 1.17183660885960112768389896766, 2.92164610838190694156295353929, 4.14012345266273986139157800617, 5.17199048767515750986656396094, 5.46872880149705668379516677869, 7.31243794275467106291144133214, 8.379117140766104693782528785094, 9.400582819765796827656643662523, 10.16407984283087259823737147151, 11.24665761829711362926886395447

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