Properties

Label 2-1045-1045.949-c0-0-7
Degree $2$
Conductor $1045$
Sign $0.286 + 0.958i$
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  + (−0.309 + 0.951i)4-s + (0.309 − 0.951i)5-s + (−1.80 − 0.587i)7-s + (0.809 − 0.587i)9-s + (−0.309 − 0.951i)11-s + (−0.809 − 0.587i)16-s + (0.690 − 0.951i)17-s + (0.309 + 0.951i)19-s + (0.809 + 0.587i)20-s − 1.17i·23-s + (−0.809 − 0.587i)25-s + (1.11 − 1.53i)28-s + (−1.11 + 1.53i)35-s + (0.309 + 0.951i)36-s + 1.17i·43-s + 0.999·44-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)4-s + (0.309 − 0.951i)5-s + (−1.80 − 0.587i)7-s + (0.809 − 0.587i)9-s + (−0.309 − 0.951i)11-s + (−0.809 − 0.587i)16-s + (0.690 − 0.951i)17-s + (0.309 + 0.951i)19-s + (0.809 + 0.587i)20-s − 1.17i·23-s + (−0.809 − 0.587i)25-s + (1.11 − 1.53i)28-s + (−1.11 + 1.53i)35-s + (0.309 + 0.951i)36-s + 1.17i·43-s + 0.999·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.286 + 0.958i$
Analytic conductor: \(0.521522\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ 0.286 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7712829715\)
\(L(\frac12)\) \(\approx\) \(0.7712829715\)
\(L(1)\) 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 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + 1.17iT - T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - 1.17iT - T^{2} \)
47 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.309 - 0.951i)T^{2} \)
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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

−9.676149345411763453496921535948, −9.310499830238865128966878857218, −8.352817926728751171290607575914, −7.48408156880170114600864382361, −6.61250716204408212517590661979, −5.76535839848531155309879708630, −4.48715950418275564933690547249, −3.65237288046653406491478526048, −2.90900610716713555444577373122, −0.72978542713341857223768524333, 1.86155229764754625425459855511, 2.93456754162108800685458813777, 4.07639387084559780013526063422, 5.38092857440446642405383974307, 6.01384794047537770724602692529, 6.89080000509376905665192877996, 7.46495416281157664474304657515, 9.063946692326601879966763442585, 9.635298039313261951985931091317, 10.23938054066813213299732296273

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