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Properties

Label 2-420-105.104-c1-0-8
Degree $2$
Conductor $420$
Sign $0.924 + 0.381i$
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  + (−1.41 + i)3-s + 2.23·5-s + (−1.41 − 2.23i)7-s + (1.00 − 2.82i)9-s − 2.82i·11-s + 2.82·13-s + (−3.16 + 2.23i)15-s − 4i·17-s + 6.32i·19-s + (4.23 + 1.74i)21-s + 6.32·23-s + 5.00·25-s + (1.41 + 5.00i)27-s − 5.65i·29-s − 6.32i·31-s + ⋯
L(s)  = 1  + (−0.816 + 0.577i)3-s + 0.999·5-s + (−0.534 − 0.845i)7-s + (0.333 − 0.942i)9-s − 0.852i·11-s + 0.784·13-s + (−0.816 + 0.577i)15-s − 0.970i·17-s + 1.45i·19-s + (0.924 + 0.381i)21-s + 1.31·23-s + 1.00·25-s + (0.272 + 0.962i)27-s − 1.05i·29-s − 1.13i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.924 + 0.381i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.924 + 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18753 - 0.235396i\)
\(L(\frac12)\) \(\approx\) \(1.18753 - 0.235396i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.41 - i)T \)
5 \( 1 - 2.23T \)
7 \( 1 + (1.41 + 2.23i)T \)
good11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 6.32iT - 19T^{2} \)
23 \( 1 - 6.32T + 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + 6.32iT - 31T^{2} \)
37 \( 1 - 8.94iT - 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 4.47iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 6.32T + 53T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 4.47iT - 67T^{2} \)
71 \( 1 - 14.1iT - 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 10iT - 83T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 + 8.48T + 97T^{2} \)
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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

−11.01803978044560497291365491012, −10.19496150527712912404238153999, −9.653621650232642840069304413176, −8.628279865993371263078728992206, −7.15155894166557288599156578353, −6.17285268590498252192915436040, −5.59268364855033762182446104848, −4.28481234749642440204925890939, −3.15316147182265164759315853371, −0.987678258458286877962714640627, 1.52081472742875585170953109022, 2.79494655502286307571722410622, 4.76981989145424550035102596788, 5.65697752135152393605629247827, 6.47570285580527288029794096978, 7.19591754959350170738261242887, 8.739206358044206851978646546176, 9.367265248581611727575792874795, 10.58645755132358333965029149218, 11.09067740722982976654089474324

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