Properties

Label 2-420-60.59-c1-0-39
Degree $2$
Conductor $420$
Sign $0.805 - 0.592i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (1 − 1.41i)3-s − 2.00·4-s + (2.12 + 0.707i)5-s + (2.00 + 1.41i)6-s − 7-s − 2.82i·8-s + (−1.00 − 2.82i)9-s + (−1.00 + 3i)10-s + 4.24·11-s + (−2.00 + 2.82i)12-s + 6i·13-s − 1.41i·14-s + (3.12 − 2.29i)15-s + 4.00·16-s + 4.24·17-s + ⋯
L(s)  = 1  + 0.999i·2-s + (0.577 − 0.816i)3-s − 1.00·4-s + (0.948 + 0.316i)5-s + (0.816 + 0.577i)6-s − 0.377·7-s − 1.00i·8-s + (−0.333 − 0.942i)9-s + (−0.316 + 0.948i)10-s + 1.27·11-s + (−0.577 + 0.816i)12-s + 1.66i·13-s − 0.377i·14-s + (0.805 − 0.592i)15-s + 1.00·16-s + 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68061 + 0.550944i\)
\(L(\frac12)\) \(\approx\) \(1.68061 + 0.550944i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (-1 + 1.41i)T \)
5 \( 1 + (-2.12 - 0.707i)T \)
7 \( 1 + T \)
good11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 1.41iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 + 8.48T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + 7.07iT - 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.49420100682111351303541289800, −9.803896962976699937419540744040, −9.301342999732021013447176717297, −8.613719995279780514130370317310, −7.27446409638868698896639295614, −6.60062329651375265125902977713, −6.12327881386186657434668822021, −4.56475132476954870516132609969, −3.18816778031188797535248616826, −1.50086474768739599886562388192, 1.51878656302214268501630434690, 3.04284848597830201678830330889, 3.80502212138265729756277375543, 5.18327689751781217457037923577, 5.90706803593045782423882382444, 7.85954483906425067517842870383, 8.746780383321675469890364966786, 9.590619144181829893583929188922, 10.09474035331702418206152297915, 10.76694752139929866734345928182

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