Properties

Label 2-1045-1.1-c1-0-45
Degree $2$
Conductor $1045$
Sign $-1$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.412·2-s + 1.67·3-s − 1.83·4-s − 5-s − 0.691·6-s + 0.0703·7-s + 1.57·8-s − 0.184·9-s + 0.412·10-s − 11-s − 3.07·12-s + 1.09·13-s − 0.0289·14-s − 1.67·15-s + 3.00·16-s − 2.37·17-s + 0.0758·18-s − 19-s + 1.83·20-s + 0.117·21-s + 0.412·22-s − 2.83·23-s + 2.64·24-s + 25-s − 0.449·26-s − 5.34·27-s − 0.128·28-s + ⋯
L(s)  = 1  − 0.291·2-s + 0.968·3-s − 0.915·4-s − 0.447·5-s − 0.282·6-s + 0.0265·7-s + 0.558·8-s − 0.0613·9-s + 0.130·10-s − 0.301·11-s − 0.886·12-s + 0.302·13-s − 0.00774·14-s − 0.433·15-s + 0.752·16-s − 0.577·17-s + 0.0178·18-s − 0.229·19-s + 0.409·20-s + 0.0257·21-s + 0.0878·22-s − 0.590·23-s + 0.540·24-s + 0.200·25-s − 0.0881·26-s − 1.02·27-s − 0.0243·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-1$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
11 \( 1 + T \)
19 \( 1 + T \)
good2 \( 1 + 0.412T + 2T^{2} \)
3 \( 1 - 1.67T + 3T^{2} \)
7 \( 1 - 0.0703T + 7T^{2} \)
13 \( 1 - 1.09T + 13T^{2} \)
17 \( 1 + 2.37T + 17T^{2} \)
23 \( 1 + 2.83T + 23T^{2} \)
29 \( 1 + 9.85T + 29T^{2} \)
31 \( 1 - 1.51T + 31T^{2} \)
37 \( 1 + 1.38T + 37T^{2} \)
41 \( 1 - 2.59T + 41T^{2} \)
43 \( 1 + 5.63T + 43T^{2} \)
47 \( 1 + 4.95T + 47T^{2} \)
53 \( 1 + 3.23T + 53T^{2} \)
59 \( 1 - 5.63T + 59T^{2} \)
61 \( 1 - 1.13T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + 3.37T + 71T^{2} \)
73 \( 1 + 4.54T + 73T^{2} \)
79 \( 1 - 2.22T + 79T^{2} \)
83 \( 1 - 2.48T + 83T^{2} \)
89 \( 1 - 6.45T + 89T^{2} \)
97 \( 1 - 9.43T + 97T^{2} \)
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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

−9.285585871062046068719510638172, −8.685692888397283206738086163140, −8.048044699165797124862163227704, −7.41573262184427871949513638976, −6.07334005372891380115899414012, −4.98574385124782228315384122898, −4.00307030489092691870585099779, −3.25184117695079199248778051745, −1.88449877567300572374722602720, 0, 1.88449877567300572374722602720, 3.25184117695079199248778051745, 4.00307030489092691870585099779, 4.98574385124782228315384122898, 6.07334005372891380115899414012, 7.41573262184427871949513638976, 8.048044699165797124862163227704, 8.685692888397283206738086163140, 9.285585871062046068719510638172

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