Properties

Label 2-1045-1045.284-c0-0-0
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

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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: $\chi_{1045} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ -0.550 + 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4965845369\)
\(L(\frac12)\) \(\approx\) \(0.4965845369\)
\(L(1)\) 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 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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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

−10.46928681669208622789042408856, −9.674295193824435293843026368980, −8.871427741003624023628505384389, −8.076890547637702730937032525175, −7.15355582849844669644080438137, −6.47137403209915347191360585446, −5.88313498391416528075764014017, −5.00526866480211695744943522114, −3.89082791983339165068547208944, −2.26130537906733152861242861994, 0.57402276687969169064111777264, 1.74387819764198111530161953332, 2.86948013701621776684247653956, 4.08275180719986615025978957516, 5.04628685002785603161654090759, 6.03516447731750310650178453806, 7.18013880145692810304374731024, 8.336973893399868681911700609163, 8.993411325280886047778675526979, 9.781945118835883919666537851535

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