Properties

Label 2-1045-1.1-c3-0-169
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  + 3.59·2-s + 4.02·3-s + 4.91·4-s − 5·5-s + 14.4·6-s + 10.5·7-s − 11.0·8-s − 10.8·9-s − 17.9·10-s − 11·11-s + 19.7·12-s − 33.8·13-s + 37.9·14-s − 20.1·15-s − 79.1·16-s + 70.8·17-s − 38.8·18-s + 19·19-s − 24.5·20-s + 42.4·21-s − 39.5·22-s − 83.8·23-s − 44.5·24-s + 25·25-s − 121.·26-s − 152.·27-s + 51.8·28-s + ⋯
L(s)  = 1  + 1.27·2-s + 0.774·3-s + 0.614·4-s − 0.447·5-s + 0.983·6-s + 0.569·7-s − 0.489·8-s − 0.400·9-s − 0.568·10-s − 0.301·11-s + 0.476·12-s − 0.723·13-s + 0.723·14-s − 0.346·15-s − 1.23·16-s + 1.01·17-s − 0.509·18-s + 0.229·19-s − 0.274·20-s + 0.440·21-s − 0.383·22-s − 0.760·23-s − 0.378·24-s + 0.200·25-s − 0.918·26-s − 1.08·27-s + 0.350·28-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 \( 1 - 3.59T + 8T^{2} \)
3 \( 1 - 4.02T + 27T^{2} \)
7 \( 1 - 10.5T + 343T^{2} \)
13 \( 1 + 33.8T + 2.19e3T^{2} \)
17 \( 1 - 70.8T + 4.91e3T^{2} \)
23 \( 1 + 83.8T + 1.21e4T^{2} \)
29 \( 1 + 121.T + 2.43e4T^{2} \)
31 \( 1 + 46.6T + 2.97e4T^{2} \)
37 \( 1 - 180.T + 5.06e4T^{2} \)
41 \( 1 + 236.T + 6.89e4T^{2} \)
43 \( 1 + 471.T + 7.95e4T^{2} \)
47 \( 1 - 137.T + 1.03e5T^{2} \)
53 \( 1 + 491.T + 1.48e5T^{2} \)
59 \( 1 + 294.T + 2.05e5T^{2} \)
61 \( 1 + 303.T + 2.26e5T^{2} \)
67 \( 1 + 466.T + 3.00e5T^{2} \)
71 \( 1 + 266.T + 3.57e5T^{2} \)
73 \( 1 - 1.03e3T + 3.89e5T^{2} \)
79 \( 1 - 1.17e3T + 4.93e5T^{2} \)
83 \( 1 + 360.T + 5.71e5T^{2} \)
89 \( 1 - 988.T + 7.04e5T^{2} \)
97 \( 1 - 1.29e3T + 9.12e5T^{2} \)
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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

−9.082115013206114138022916282659, −8.059875383211667944033440161310, −7.63679957356465698474661188979, −6.35725160416043368409571564319, −5.38189398328714208104569470046, −4.74751424151223907542951438117, −3.64888098971297665868624790982, −3.05296914842557405683161262102, −1.95678432150343602037035139912, 0, 1.95678432150343602037035139912, 3.05296914842557405683161262102, 3.64888098971297665868624790982, 4.74751424151223907542951438117, 5.38189398328714208104569470046, 6.35725160416043368409571564319, 7.63679957356465698474661188979, 8.059875383211667944033440161310, 9.082115013206114138022916282659

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