↑

Properties

Label 2-1045-1045.379-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  + (0.5 + 1.53i)2-s + (0.5 + 0.363i)3-s + (−1.30 + 0.951i)4-s + (0.309 − 0.951i)5-s + (−0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.190 − 0.587i)9-s + 1.61·10-s + (0.309 + 0.951i)11-s − 12-s + (0.5 + 1.53i)13-s + (0.5 − 0.363i)15-s + (0.809 − 0.587i)18-s + (−0.809 − 0.587i)19-s + (0.499 + 1.53i)20-s + ⋯
L(s)  = 1  + (0.5 + 1.53i)2-s + (0.5 + 0.363i)3-s + (−1.30 + 0.951i)4-s + (0.309 − 0.951i)5-s + (−0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.190 − 0.587i)9-s + 1.61·10-s + (0.309 + 0.951i)11-s − 12-s + (0.5 + 1.53i)13-s + (0.5 − 0.363i)15-s + (0.809 − 0.587i)18-s + (−0.809 − 0.587i)19-s + (0.499 + 1.53i)20-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: $\chi_{1045} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ -0.550 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.537461622\)
\(L(\frac12)\) \(\approx\) \(1.537461622\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
3 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + 0.618T + T^{2} \)
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 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
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

−9.913327137584056216275044313444, −9.122324069068920243535104805099, −8.784746921067783969865441658555, −7.920692742783966456935592639484, −6.75120186881800275222484995744, −6.38763345195463496836028823880, −5.23043897242219288186862411106, −4.39528502742347894023492162679, −3.89892406618880721670914432543, −1.97652122381475881612918134438, 1.43910140425464993463648379690, 2.68617955853740078419148178268, 3.09260618057226455113250156883, 4.09528041914357984902701638935, 5.46310547195382735406388250117, 6.18471697891313403346851608430, 7.52390669647771361321291684439, 8.282547030202103697690659654807, 9.237238184792130192840114501027, 10.32596393985408740355266241175

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