↑

Properties

Label 2-420-105.2-c1-0-1
Degree $2$
Conductor $420$
Sign $-0.990 - 0.137i$
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  + (−0.795 + 1.53i)3-s + (0.746 + 2.10i)5-s + (−2.40 + 1.10i)7-s + (−1.73 − 2.44i)9-s + (−2.63 + 1.52i)11-s + (0.512 − 0.512i)13-s + (−3.83 − 0.529i)15-s + (3.88 + 1.04i)17-s + (−5.78 − 3.34i)19-s + (0.219 − 4.57i)21-s + (2.29 − 0.614i)23-s + (−3.88 + 3.14i)25-s + (5.14 − 0.719i)27-s − 8.51·29-s + (0.921 + 1.59i)31-s + ⋯
L(s)  = 1  + (−0.459 + 0.888i)3-s + (0.333 + 0.942i)5-s + (−0.909 + 0.416i)7-s + (−0.577 − 0.816i)9-s + (−0.795 + 0.459i)11-s + (0.142 − 0.142i)13-s + (−0.990 − 0.136i)15-s + (0.943 + 0.252i)17-s + (−1.32 − 0.766i)19-s + (0.0479 − 0.998i)21-s + (0.478 − 0.128i)23-s + (−0.777 + 0.629i)25-s + (0.990 − 0.138i)27-s − 1.58·29-s + (0.165 + 0.286i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.990 - 0.137i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.990 - 0.137i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0469402 + 0.679261i\)
\(L(\frac12)\) \(\approx\) \(0.0469402 + 0.679261i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.795 - 1.53i)T \)
5 \( 1 + (-0.746 - 2.10i)T \)
7 \( 1 + (2.40 - 1.10i)T \)
good11 \( 1 + (2.63 - 1.52i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.512 + 0.512i)T - 13iT^{2} \)
17 \( 1 + (-3.88 - 1.04i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.78 + 3.34i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.29 + 0.614i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 8.51T + 29T^{2} \)
31 \( 1 + (-0.921 - 1.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.78 + 0.747i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 8.91iT - 41T^{2} \)
43 \( 1 + (7.54 - 7.54i)T - 43iT^{2} \)
47 \( 1 + (-1.95 - 7.28i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.81 - 10.5i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.06 + 1.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.12 + 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.03 + 3.86i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.41iT - 71T^{2} \)
73 \( 1 + (-11.0 - 2.95i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-12.1 - 7.01i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.995 - 0.995i)T + 83iT^{2} \)
89 \( 1 + (-0.706 + 1.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.556 + 0.556i)T + 97iT^{2} \)
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

−11.26870149497625358196566396808, −10.72527144284547166175334185548, −9.839326062861267523381760245179, −9.329322153530052098435538216350, −7.964631694870321493440389969323, −6.65913437123828591256891632429, −6.00229212714045509507341108280, −4.94953964457307318010115807478, −3.55828724234166297981299831439, −2.61755118207104185959349633701, 0.43747928650082175225281982626, 2.05152978646010583022846438761, 3.70139782175422557409594574265, 5.27530778498840837947084234546, 5.90990128446432234738422998816, 6.98245949086396431568259485827, 7.974297203219330729255291875507, 8.809350080430560417743834549145, 9.964903928980049326606506622758, 10.74023298019716743019869174813

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