↑

Properties

Label 2-42e2-63.58-c1-0-20
Degree $2$
Conductor $1764$
Sign $0.264 - 0.964i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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  + (1.71 + 0.272i)3-s + (1.97 + 3.41i)5-s + (2.85 + 0.931i)9-s + (−0.471 + 0.816i)11-s + (0.5 − 0.866i)13-s + (2.44 + 6.37i)15-s + (−2.80 − 4.85i)17-s + (−0.641 + 1.11i)19-s + (2.33 + 4.03i)23-s + (−5.27 + 9.13i)25-s + (4.62 + 2.36i)27-s + (3.83 + 6.63i)29-s + 7.82·31-s + (−1.02 + 1.26i)33-s + (4.91 − 8.51i)37-s + ⋯
L(s)  = 1  + (0.987 + 0.157i)3-s + (0.881 + 1.52i)5-s + (0.950 + 0.310i)9-s + (−0.142 + 0.246i)11-s + (0.138 − 0.240i)13-s + (0.630 + 1.64i)15-s + (−0.679 − 1.17i)17-s + (−0.147 + 0.254i)19-s + (0.485 + 0.841i)23-s + (−1.05 + 1.82i)25-s + (0.890 + 0.455i)27-s + (0.711 + 1.23i)29-s + 1.40·31-s + (−0.179 + 0.220i)33-s + (0.807 − 1.39i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.264 - 0.964i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.264 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.938525818\)
\(L(\frac12)\) \(\approx\) \(2.938525818\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.272i)T \)
7 \( 1 \)
good5 \( 1 + (-1.97 - 3.41i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.471 - 0.816i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.80 + 4.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.641 - 1.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.33 - 4.03i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.83 - 6.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.82T + 31T^{2} \)
37 \( 1 + (-4.91 + 8.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.471 - 0.816i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.63 + 8.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.28T + 47T^{2} \)
53 \( 1 + (-4.61 - 7.98i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.54T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + 1.71T + 67T^{2} \)
71 \( 1 - 3.54T + 71T^{2} \)
73 \( 1 + (1.35 + 2.35i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 + (-0.198 - 0.343i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.75 - 4.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.77 - 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.457566673809053553107602489543, −8.871188593956576978054017770527, −7.73357380635481408480868693956, −7.10924787357349049603202451463, −6.50706249349935733645185738458, −5.44179997077690250333329468725, −4.38145738576071631338028231959, −3.12207293149105791265284268240, −2.75974393106094992001693458586, −1.70599449384616867470865433562, 1.02916829310921066234082343416, 1.95586876646729116764619923163, 2.95735808390867844618049009144, 4.46161586427140684065171399466, 4.64362560509579329397094439077, 6.10170323693565694505947386830, 6.53944383958620719017745254596, 8.030609859388674172523467551123, 8.408406966404118175166934286067, 8.950511026349469123664611560836

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