Properties

Label 2-1045-1.1-c5-0-218
Degree $2$
Conductor $1045$
Sign $-1$
Analytic cond. $167.601$
Root an. cond. $12.9460$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11.2·2-s + 20.0·3-s + 95.1·4-s − 25·5-s − 226.·6-s − 33.0·7-s − 712.·8-s + 160.·9-s + 281.·10-s + 121·11-s + 1.91e3·12-s + 134.·13-s + 372.·14-s − 502.·15-s + 4.98e3·16-s − 121.·17-s − 1.80e3·18-s + 361·19-s − 2.37e3·20-s − 663.·21-s − 1.36e3·22-s − 163.·23-s − 1.43e4·24-s + 625·25-s − 1.51e3·26-s − 1.66e3·27-s − 3.14e3·28-s + ⋯
L(s)  = 1  − 1.99·2-s + 1.28·3-s + 2.97·4-s − 0.447·5-s − 2.56·6-s − 0.255·7-s − 3.93·8-s + 0.659·9-s + 0.891·10-s + 0.301·11-s + 3.83·12-s + 0.220·13-s + 0.508·14-s − 0.576·15-s + 4.87·16-s − 0.102·17-s − 1.31·18-s + 0.229·19-s − 1.32·20-s − 0.328·21-s − 0.601·22-s − 0.0645·23-s − 5.06·24-s + 0.200·25-s − 0.439·26-s − 0.438·27-s − 0.758·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-1$
Analytic conductor: \(167.601\)
Root analytic conductor: \(12.9460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1045,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 25T \)
11 \( 1 - 121T \)
19 \( 1 - 361T \)
good2 \( 1 + 11.2T + 32T^{2} \)
3 \( 1 - 20.0T + 243T^{2} \)
7 \( 1 + 33.0T + 1.68e4T^{2} \)
13 \( 1 - 134.T + 3.71e5T^{2} \)
17 \( 1 + 121.T + 1.41e6T^{2} \)
23 \( 1 + 163.T + 6.43e6T^{2} \)
29 \( 1 + 3.85e3T + 2.05e7T^{2} \)
31 \( 1 - 447.T + 2.86e7T^{2} \)
37 \( 1 - 8.45e3T + 6.93e7T^{2} \)
41 \( 1 + 1.21e4T + 1.15e8T^{2} \)
43 \( 1 - 1.72e4T + 1.47e8T^{2} \)
47 \( 1 - 2.27e3T + 2.29e8T^{2} \)
53 \( 1 - 3.30e4T + 4.18e8T^{2} \)
59 \( 1 + 345.T + 7.14e8T^{2} \)
61 \( 1 + 2.31e4T + 8.44e8T^{2} \)
67 \( 1 - 7.50e3T + 1.35e9T^{2} \)
71 \( 1 + 3.88e3T + 1.80e9T^{2} \)
73 \( 1 + 5.22e4T + 2.07e9T^{2} \)
79 \( 1 - 9.27e4T + 3.07e9T^{2} \)
83 \( 1 - 1.42e3T + 3.93e9T^{2} \)
89 \( 1 + 4.00e4T + 5.58e9T^{2} \)
97 \( 1 + 6.28e3T + 8.58e9T^{2} \)
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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

−8.861345484328292510707625644322, −8.098816164421361663638355592869, −7.56874452425790208582064177350, −6.79910591312267881382437313776, −5.80660325398251061545984102296, −3.80317952269301781130475587220, −2.96287570118016142985261262597, −2.15194211847332728825437438003, −1.13850685748034287136003426819, 0, 1.13850685748034287136003426819, 2.15194211847332728825437438003, 2.96287570118016142985261262597, 3.80317952269301781130475587220, 5.80660325398251061545984102296, 6.79910591312267881382437313776, 7.56874452425790208582064177350, 8.098816164421361663638355592869, 8.861345484328292510707625644322

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