Properties

Label 2-1045-1045.379-c0-0-4
Degree $2$
Conductor $1045$
Sign $0.964 - 0.265i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)4-s + (0.309 + 0.951i)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 + (0.809 + 0.587i)20-s + 1.90i·23-s + (−0.809 + 0.587i)25-s + (1.11 + 0.363i)28-s + (−0.690 + 0.951i)35-s + (−0.809 − 0.587i)36-s − 1.90i·43-s + 44-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)4-s + (0.309 + 0.951i)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 + (0.809 + 0.587i)20-s + 1.90i·23-s + (−0.809 + 0.587i)25-s + (1.11 + 0.363i)28-s + (−0.690 + 0.951i)35-s + (−0.809 − 0.587i)36-s − 1.90i·43-s + 44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.964 - 0.265i$
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: \((2,\ 1045,\ (\ :0),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.328262453\)
\(L(\frac12)\) \(\approx\) \(1.328262453\)
\(L(1)\) 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 \( 1 + (-0.809 - 0.587i)T \)
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 \( 1 + (-0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.21113054886577532933995941339, −9.246522579748464136010862408122, −8.825231824530437302729473624046, −7.25295890388695595125625171485, −6.81245314741722414884067953495, −6.02246635816760699225180128623, −5.21953703049033772856713006860, −3.83684259835961469185248358774, −2.52399645412985078657528973218, −1.87194181839078801009391574155, 1.53864732888383593462439974632, 2.52164043880825133351640873521, 4.22382880013617333151397361241, 4.50273962629193106970797731922, 6.03483358937890099689493189535, 6.62799677270334789006967904469, 7.81440300553720821496164026952, 8.364660757354038905880500818732, 8.945204764197273591001717889676, 10.38325839956642161863305254780

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