↑

Properties

Label 2-420-35.12-c1-0-7
Degree $2$
Conductor $420$
Sign $-0.618 + 0.785i$
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  + (0.258 − 0.965i)3-s + (−0.0833 − 2.23i)5-s + (−1.70 − 2.02i)7-s + (−0.866 − 0.499i)9-s + (1.21 + 2.10i)11-s + (−0.728 − 0.728i)13-s + (−2.17 − 0.497i)15-s + (−7.33 − 1.96i)17-s + (1.84 − 3.20i)19-s + (−2.39 + 1.12i)21-s + (0.759 + 2.83i)23-s + (−4.98 + 0.372i)25-s + (−0.707 + 0.707i)27-s − 1.99i·29-s + (6.02 − 3.47i)31-s + ⋯
L(s)  = 1  + (0.149 − 0.557i)3-s + (−0.0372 − 0.999i)5-s + (−0.645 − 0.763i)7-s + (−0.288 − 0.166i)9-s + (0.366 + 0.634i)11-s + (−0.202 − 0.202i)13-s + (−0.562 − 0.128i)15-s + (−1.77 − 0.476i)17-s + (0.424 − 0.734i)19-s + (−0.522 + 0.245i)21-s + (0.158 + 0.591i)23-s + (−0.997 + 0.0744i)25-s + (−0.136 + 0.136i)27-s − 0.369i·29-s + (1.08 − 0.624i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.618 + 0.785i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.618 + 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.481654 - 0.991859i\)
\(L(\frac12)\) \(\approx\) \(0.481654 - 0.991859i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (0.0833 + 2.23i)T \)
7 \( 1 + (1.70 + 2.02i)T \)
good11 \( 1 + (-1.21 - 2.10i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.728 + 0.728i)T + 13iT^{2} \)
17 \( 1 + (7.33 + 1.96i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.84 + 3.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.759 - 2.83i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 1.99iT - 29T^{2} \)
31 \( 1 + (-6.02 + 3.47i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.22 + 2.20i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 1.08iT - 41T^{2} \)
43 \( 1 + (-5.91 + 5.91i)T - 43iT^{2} \)
47 \( 1 + (1.28 + 4.81i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.56 + 1.76i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.39 + 2.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.217 - 0.125i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.08 - 7.79i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 + (-1.73 + 6.46i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.54 - 5.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.90 - 9.90i)T + 83iT^{2} \)
89 \( 1 + (5.81 - 10.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.8 + 10.8i)T - 97iT^{2} \)
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

−11.02686535086504355187095165396, −9.639453950020630806474789412201, −9.224963662615742366522954421134, −8.045706031994169843638073763169, −7.14837836167196402391958484278, −6.32563914099579466128735312768, −4.91491397833520659474517088877, −4.00650986671835911535352760760, −2.36288587093219284576383643762, −0.68544944132114265880464960655, 2.44102482824447755151523038197, 3.38492446334383331538302296029, 4.58200556135619131579556947230, 6.11811131470839935708491444435, 6.53245790792091133777344878536, 7.961268968458748120034227932970, 8.953338565362285582204830652348, 9.678231429999326022154683933604, 10.68334693282718816596227221191, 11.30830301771009971418343457740

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