Properties

Label 2-1045-1.1-c3-0-71
Degree $2$
Conductor $1045$
Sign $-1$
Analytic cond. $61.6569$
Root an. cond. $7.85219$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.44·2-s − 2.04·3-s + 11.7·4-s − 5·5-s + 9.06·6-s − 16.0·7-s − 16.5·8-s − 22.8·9-s + 22.2·10-s + 11·11-s − 23.9·12-s − 2.33·13-s + 71.4·14-s + 10.2·15-s − 20.2·16-s − 20.5·17-s + 101.·18-s − 19·19-s − 58.6·20-s + 32.8·21-s − 48.8·22-s + 134.·23-s + 33.7·24-s + 25·25-s + 10.3·26-s + 101.·27-s − 188.·28-s + ⋯
L(s)  = 1  − 1.57·2-s − 0.392·3-s + 1.46·4-s − 0.447·5-s + 0.616·6-s − 0.869·7-s − 0.731·8-s − 0.845·9-s + 0.702·10-s + 0.301·11-s − 0.575·12-s − 0.0497·13-s + 1.36·14-s + 0.175·15-s − 0.316·16-s − 0.292·17-s + 1.32·18-s − 0.229·19-s − 0.655·20-s + 0.341·21-s − 0.473·22-s + 1.22·23-s + 0.287·24-s + 0.200·25-s + 0.0780·26-s + 0.724·27-s − 1.27·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-1$
Analytic conductor: \(61.6569\)
Root analytic conductor: \(7.85219\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1045,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5T \)
11 \( 1 - 11T \)
19 \( 1 + 19T \)
good2 \( 1 + 4.44T + 8T^{2} \)
3 \( 1 + 2.04T + 27T^{2} \)
7 \( 1 + 16.0T + 343T^{2} \)
13 \( 1 + 2.33T + 2.19e3T^{2} \)
17 \( 1 + 20.5T + 4.91e3T^{2} \)
23 \( 1 - 134.T + 1.21e4T^{2} \)
29 \( 1 + 229.T + 2.43e4T^{2} \)
31 \( 1 - 151.T + 2.97e4T^{2} \)
37 \( 1 - 87.5T + 5.06e4T^{2} \)
41 \( 1 - 361.T + 6.89e4T^{2} \)
43 \( 1 + 267.T + 7.95e4T^{2} \)
47 \( 1 - 95.5T + 1.03e5T^{2} \)
53 \( 1 - 33.9T + 1.48e5T^{2} \)
59 \( 1 + 618.T + 2.05e5T^{2} \)
61 \( 1 + 148.T + 2.26e5T^{2} \)
67 \( 1 - 672.T + 3.00e5T^{2} \)
71 \( 1 - 287.T + 3.57e5T^{2} \)
73 \( 1 - 696.T + 3.89e5T^{2} \)
79 \( 1 - 1.18e3T + 4.93e5T^{2} \)
83 \( 1 - 599.T + 5.71e5T^{2} \)
89 \( 1 + 679.T + 7.04e5T^{2} \)
97 \( 1 - 1.39e3T + 9.12e5T^{2} \)
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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

−9.219054855495859240607520027761, −8.437048217435447788619483517680, −7.63460603140447367057519761700, −6.75917084029033515512534824544, −6.11565059041645599376258241571, −4.84112337714649127728404159173, −3.46651671365808804642587080585, −2.38289110429017147760418681760, −0.885421156364398064204302182306, 0, 0.885421156364398064204302182306, 2.38289110429017147760418681760, 3.46651671365808804642587080585, 4.84112337714649127728404159173, 6.11565059041645599376258241571, 6.75917084029033515512534824544, 7.63460603140447367057519761700, 8.437048217435447788619483517680, 9.219054855495859240607520027761

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