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Properties

Label 2-1045-1045.18-c0-0-1
Degree $2$
Conductor $1045$
Sign $0.999 - 0.00521i$
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  + (−0.951 + 0.309i)4-s + (−0.809 − 0.587i)5-s + (−0.412 + 0.809i)7-s + (0.587 − 0.809i)9-s + (0.951 + 0.309i)11-s + (0.809 − 0.587i)16-s + (0.309 + 0.0489i)17-s + (0.309 − 0.951i)19-s + (0.951 + 0.309i)20-s + (1.39 + 1.39i)23-s + (0.309 + 0.951i)25-s + (0.142 − 0.896i)28-s + (0.809 − 0.412i)35-s + (−0.309 + 0.951i)36-s + (1.39 − 1.39i)43-s − 0.999·44-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)4-s + (−0.809 − 0.587i)5-s + (−0.412 + 0.809i)7-s + (0.587 − 0.809i)9-s + (0.951 + 0.309i)11-s + (0.809 − 0.587i)16-s + (0.309 + 0.0489i)17-s + (0.309 − 0.951i)19-s + (0.951 + 0.309i)20-s + (1.39 + 1.39i)23-s + (0.309 + 0.951i)25-s + (0.142 − 0.896i)28-s + (0.809 − 0.412i)35-s + (−0.309 + 0.951i)36-s + (1.39 − 1.39i)43-s − 0.999·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.999 - 0.00521i$
Analytic conductor: \(0.521522\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ 0.999 - 0.00521i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7729411223\)
\(L(\frac12)\) \(\approx\) \(0.7729411223\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.951 - 0.309i)T^{2} \)
3 \( 1 + (-0.587 + 0.809i)T^{2} \)
7 \( 1 + (0.412 - 0.809i)T + (-0.587 - 0.809i)T^{2} \)
13 \( 1 + (-0.951 + 0.309i)T^{2} \)
17 \( 1 + (-0.309 - 0.0489i)T + (0.951 + 0.309i)T^{2} \)
23 \( 1 + (-1.39 - 1.39i)T + iT^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.587 - 0.809i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
47 \( 1 + (-0.142 - 0.278i)T + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (1.26 + 0.642i)T + (0.587 + 0.809i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.951 + 0.309i)T^{2} \)
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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

−9.644018573043420392516782739501, −9.173261199165946471950878700859, −8.833397324869548921212838042934, −7.61959196139531982401677054305, −6.94975516831909151187990710385, −5.68304010890710311628479096899, −4.78612370971722559583300362321, −3.91821926641694518179665960016, −3.17257571266069514786330980794, −1.08352969010111989399973448756, 1.09912894912133402956015690423, 3.08204040309303206533053353630, 4.08983570414524206480266263171, 4.57158209214929998934540311760, 5.89123289302616831534584278654, 6.89189358062060849374048005709, 7.60243316730959890571165886077, 8.456535992269366164649494210437, 9.326425827712064534746158343783, 10.32711186640419800135820096112

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