↑

Properties

Label 2-456-456.107-c0-0-1
Degree $2$
Conductor $456$
Sign $-0.0977 + 0.995i$
Analytic cond. $0.227573$
Root an. cond. $0.477046$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s − 3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)6-s + 0.999·8-s + 9-s − 1.73i·11-s + (0.499 − 0.866i)12-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + (−1.49 + 0.866i)22-s − 0.999·24-s + (0.5 − 0.866i)25-s − 27-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s − 3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)6-s + 0.999·8-s + 9-s − 1.73i·11-s + (0.499 − 0.866i)12-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + (−1.49 + 0.866i)22-s − 0.999·24-s + (0.5 − 0.866i)25-s − 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $-0.0977 + 0.995i$
Analytic conductor: \(0.227573\)
Root analytic conductor: \(0.477046\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :0),\ -0.0977 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4728155065\)
\(L(\frac12)\) \(\approx\) \(0.4728155065\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - 1.73iT - T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
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

−11.01379146998844273809687118425, −10.48922595861692886782449338682, −9.390166381542645990424770576063, −8.548806518050704048874077564861, −7.48824038667504007446512576961, −6.34496389577891735348173327770, −5.25645534924896629231851122890, −4.10285901389674550174724170390, −2.83414689458272864381196630463, −0.927576842515372413123751675430, 1.63238574284247379840390154843, 4.18974730864581540222376691032, 5.10301497497721518756006403472, 5.94960751158846338744690232339, 7.11511576935932487986078170243, 7.44915838408115270125216898996, 8.860350299159092701915937093672, 9.906952516484655003577807188418, 10.28927775703946148463245526615, 11.43158271047029854928625033063

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