↑

Properties

Label 2-1045-1045.284-c0-0-3
Degree $2$
Conductor $1045$
Sign $0.550 - 0.835i$
Analytic cond. $0.521522$
Root an. cond. $0.722165$
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  + (0.809 + 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.690 + 0.951i)7-s + (−0.309 + 0.951i)9-s + (0.809 − 0.587i)11-s + (0.309 + 0.951i)16-s + (1.80 − 0.587i)17-s + (−0.809 + 0.587i)19-s + (−0.309 − 0.951i)20-s + 1.90i·23-s + (0.309 + 0.951i)25-s + (−1.11 + 0.363i)28-s + (1.11 − 0.363i)35-s + (−0.809 + 0.587i)36-s − 1.90i·43-s + 44-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.690 + 0.951i)7-s + (−0.309 + 0.951i)9-s + (0.809 − 0.587i)11-s + (0.309 + 0.951i)16-s + (1.80 − 0.587i)17-s + (−0.809 + 0.587i)19-s + (−0.309 − 0.951i)20-s + 1.90i·23-s + (0.309 + 0.951i)25-s + (−1.11 + 0.363i)28-s + (1.11 − 0.363i)35-s + (−0.809 + 0.587i)36-s − 1.90i·43-s + 44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.550 - 0.835i$
Analytic conductor: \(0.521522\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ 0.550 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.019188591\)
\(L(\frac12)\) \(\approx\) \(1.019188591\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 - 1.90iT - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + 1.90iT - T^{2} \)
47 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T^{2} \)
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

−10.29006433694776466035436344851, −9.267961050873953237039146212036, −8.451422156648204608913944644561, −7.84435562097883528455274808517, −7.06375358034902341884526212948, −5.88618136283352705684589311026, −5.28984435768582706539144319850, −3.70414988938447281004788445922, −3.19237767101472123263396280622, −1.79350393639726301672892269740, 1.01794471533447493018936847108, 2.79806462571601789785699824968, 3.63249681725534015376406021571, 4.56323279759636789708873396419, 6.32221189453030459829490628738, 6.41811534395533462293071167400, 7.30210656246040789630173681309, 8.155437872124007957822726027107, 9.410201916291001669687414655901, 10.13856885324067918274051365380

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