Properties

Label 2-1045-1.1-c3-0-176
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  + 3.55·2-s + 6.62·3-s + 4.64·4-s − 5·5-s + 23.5·6-s − 2.03·7-s − 11.9·8-s + 16.8·9-s − 17.7·10-s − 11·11-s + 30.7·12-s − 18.7·13-s − 7.22·14-s − 33.1·15-s − 79.5·16-s − 128.·17-s + 59.9·18-s + 19·19-s − 23.2·20-s − 13.4·21-s − 39.1·22-s + 68.3·23-s − 79.0·24-s + 25·25-s − 66.7·26-s − 67.0·27-s − 9.43·28-s + ⋯
L(s)  = 1  + 1.25·2-s + 1.27·3-s + 0.580·4-s − 0.447·5-s + 1.60·6-s − 0.109·7-s − 0.527·8-s + 0.624·9-s − 0.562·10-s − 0.301·11-s + 0.740·12-s − 0.400·13-s − 0.137·14-s − 0.570·15-s − 1.24·16-s − 1.83·17-s + 0.785·18-s + 0.229·19-s − 0.259·20-s − 0.139·21-s − 0.379·22-s + 0.620·23-s − 0.672·24-s + 0.200·25-s − 0.503·26-s − 0.478·27-s − 0.0636·28-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: $\chi_{1045} (1, \cdot )$
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 \( 1 - 3.55T + 8T^{2} \)
3 \( 1 - 6.62T + 27T^{2} \)
7 \( 1 + 2.03T + 343T^{2} \)
13 \( 1 + 18.7T + 2.19e3T^{2} \)
17 \( 1 + 128.T + 4.91e3T^{2} \)
23 \( 1 - 68.3T + 1.21e4T^{2} \)
29 \( 1 + 89.8T + 2.43e4T^{2} \)
31 \( 1 + 43.4T + 2.97e4T^{2} \)
37 \( 1 + 142.T + 5.06e4T^{2} \)
41 \( 1 - 367.T + 6.89e4T^{2} \)
43 \( 1 + 298.T + 7.95e4T^{2} \)
47 \( 1 + 51.7T + 1.03e5T^{2} \)
53 \( 1 - 35.2T + 1.48e5T^{2} \)
59 \( 1 - 374.T + 2.05e5T^{2} \)
61 \( 1 - 470.T + 2.26e5T^{2} \)
67 \( 1 - 682.T + 3.00e5T^{2} \)
71 \( 1 + 636.T + 3.57e5T^{2} \)
73 \( 1 + 400.T + 3.89e5T^{2} \)
79 \( 1 + 830.T + 4.93e5T^{2} \)
83 \( 1 - 1.25e3T + 5.71e5T^{2} \)
89 \( 1 + 512.T + 7.04e5T^{2} \)
97 \( 1 + 575.T + 9.12e5T^{2} \)
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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

−8.971740527039313866570056663434, −8.408926982024338646786847947868, −7.35102256575714214881340708111, −6.59929335064766863030854981985, −5.40475547119648757773564875861, −4.51347314297103953731858621374, −3.74664355706408905871465856343, −2.91064714714915531561851897104, −2.13786685480764230616142093979, 0, 2.13786685480764230616142093979, 2.91064714714915531561851897104, 3.74664355706408905871465856343, 4.51347314297103953731858621374, 5.40475547119648757773564875861, 6.59929335064766863030854981985, 7.35102256575714214881340708111, 8.408926982024338646786847947868, 8.971740527039313866570056663434

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