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Properties

Label 2-420-140.139-c1-0-43
Degree $2$
Conductor $420$
Sign $-0.992 + 0.121i$
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  + (0.846 − 1.13i)2-s + i·3-s + (−0.567 − 1.91i)4-s + (−2.22 + 0.235i)5-s + (1.13 + 0.846i)6-s + (−2.54 − 0.708i)7-s + (−2.65 − 0.980i)8-s − 9-s + (−1.61 + 2.71i)10-s − 5.27i·11-s + (1.91 − 0.567i)12-s + 2.60·13-s + (−2.96 + 2.28i)14-s + (−0.235 − 2.22i)15-s + (−3.35 + 2.17i)16-s − 3.66·17-s + ⋯
L(s)  = 1  + (0.598 − 0.801i)2-s + 0.577i·3-s + (−0.283 − 0.958i)4-s + (−0.994 + 0.105i)5-s + (0.462 + 0.345i)6-s + (−0.963 − 0.267i)7-s + (−0.938 − 0.346i)8-s − 0.333·9-s + (−0.510 + 0.859i)10-s − 1.59i·11-s + (0.553 − 0.163i)12-s + 0.722·13-s + (−0.791 + 0.611i)14-s + (−0.0608 − 0.574i)15-s + (−0.839 + 0.544i)16-s − 0.889·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.992 + 0.121i$
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: \((2,\ 420,\ (\ :1/2),\ -0.992 + 0.121i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0431800 - 0.706699i\)
\(L(\frac12)\) \(\approx\) \(0.0431800 - 0.706699i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.846 + 1.13i)T \)
3 \( 1 - iT \)
5 \( 1 + (2.22 - 0.235i)T \)
7 \( 1 + (2.54 + 0.708i)T \)
good11 \( 1 + 5.27iT - 11T^{2} \)
13 \( 1 - 2.60T + 13T^{2} \)
17 \( 1 + 3.66T + 17T^{2} \)
19 \( 1 + 6.51T + 19T^{2} \)
23 \( 1 + 3.54T + 23T^{2} \)
29 \( 1 - 6.64T + 29T^{2} \)
31 \( 1 - 4.51T + 31T^{2} \)
37 \( 1 - 0.593iT - 37T^{2} \)
41 \( 1 + 8.01iT - 41T^{2} \)
43 \( 1 - 0.678T + 43T^{2} \)
47 \( 1 + 5.79iT - 47T^{2} \)
53 \( 1 - 1.21iT - 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 11.9iT - 61T^{2} \)
67 \( 1 - 8.59T + 67T^{2} \)
71 \( 1 - 0.618iT - 71T^{2} \)
73 \( 1 - 1.27T + 73T^{2} \)
79 \( 1 + 6.11iT - 79T^{2} \)
83 \( 1 + 12.0iT - 83T^{2} \)
89 \( 1 + 15.9iT - 89T^{2} \)
97 \( 1 + 0.241T + 97T^{2} \)
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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

−10.73766300611346947728194754442, −10.35129887019137968192876125134, −8.914777987990072513895890315151, −8.438112992264507022393003633965, −6.61265806066040333615777131622, −5.94787124276317777831559727948, −4.41005629432076251343538624711, −3.74333245271301130738791490324, −2.83642426710862336629583956934, −0.35924458994744113576902000733, 2.56484427392529005803179469704, 3.96053560252906680209686696019, 4.75039555575424190247391203556, 6.45565324781915102996766941859, 6.63495938491155398412618464517, 7.86631690168336752720575832957, 8.517815274584524894528740212937, 9.582963775387161582069479021648, 10.96791485679768510092971275859, 12.11629109048472813593117143766

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