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Properties

Label 2-1045-1045.284-c0-0-1
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  + (0.363 − 1.11i)2-s + (−1.53 + 1.11i)3-s + (−0.309 − 0.224i)4-s + (0.309 + 0.951i)5-s + (0.690 + 2.12i)6-s + (0.587 − 0.427i)8-s + (0.809 − 2.48i)9-s + 1.17·10-s + (−0.309 + 0.951i)11-s + 0.726·12-s + (−0.363 + 1.11i)13-s + (−1.53 − 1.11i)15-s + (−0.381 − 1.17i)16-s + (−2.48 − 1.80i)18-s + (−0.809 + 0.587i)19-s + (0.118 − 0.363i)20-s + ⋯
L(s)  = 1  + (0.363 − 1.11i)2-s + (−1.53 + 1.11i)3-s + (−0.309 − 0.224i)4-s + (0.309 + 0.951i)5-s + (0.690 + 2.12i)6-s + (0.587 − 0.427i)8-s + (0.809 − 2.48i)9-s + 1.17·10-s + (−0.309 + 0.951i)11-s + 0.726·12-s + (−0.363 + 1.11i)13-s + (−1.53 − 1.11i)15-s + (−0.381 − 1.17i)16-s + (−2.48 − 1.80i)18-s + (−0.809 + 0.587i)19-s + (0.118 − 0.363i)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

\(L(\frac{1}{2})\) \(\approx\) \(0.7689669611\)
\(L(\frac12)\) \(\approx\) \(0.7689669611\)
\(L(1)\) 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 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + 1.90T + T^{2} \)
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 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
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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

−10.31053518884555325410716066063, −10.00122588845005154211236209778, −9.296707779434162160935512032812, −7.42776732691148505705414861613, −6.62613607839968710648849483067, −5.89498714173737366242539437264, −4.58805524940661374916649996997, −4.30330419189378730050378248913, −3.11688648341726386143399485041, −1.83058042912092744945749190543, 0.77610587510365188026763751402, 2.21343594037291468385900676210, 4.51892297915229820819994242329, 5.24900965899549107882287310215, 5.85545022317919827028073711375, 6.29208831855751287106851636928, 7.37390299856750103288006525591, 7.905681475496350599744662755325, 8.750113468009775135753521174941, 10.28307012717809227467724851462

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