↑

Properties

Label 2-456-456.107-c0-0-0
Degree $2$
Conductor $456$
Sign $-0.813 - 0.582i$
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 + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + 1.73i·11-s + (−0.499 − 0.866i)12-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (0.5 − 0.866i)19-s + (−1.49 + 0.866i)22-s + (0.499 − 0.866i)24-s + (0.5 − 0.866i)25-s + 0.999·27-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + 1.73i·11-s + (−0.499 − 0.866i)12-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (0.5 − 0.866i)19-s + (−1.49 + 0.866i)22-s + (0.499 − 0.866i)24-s + (0.5 − 0.866i)25-s + 0.999·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $-0.813 - 0.582i$
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: \((2,\ 456,\ (\ :0),\ -0.813 - 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8510636377\)
\(L(\frac12)\) \(\approx\) \(0.8510636377\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
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

−11.84708798434971949407740724291, −10.72394186087967592981814241687, −9.656431197150335986781669631512, −9.103414878346195371832364252670, −7.81010786311308769323611017373, −6.88584071533818209236956192682, −5.98043932553013673086174479937, −4.76706458767768436617974579324, −4.40748393701572650419158587144, −2.91174690276357761426582540517, 1.13274570736402288988657608237, 2.68417590642925217355310705742, 3.84821520661140768978084113995, 5.48166831707889028374776646466, 5.82468617192248558409080989563, 7.09112508615655315488550482808, 8.323493674866451858637993756225, 9.152334457947671653994766385383, 10.55515633621253543991419579612, 10.99643139811979512298205390055

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