Properties

Label 2-1045-1.1-c5-0-47
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.82·2-s − 29.9·3-s − 8.68·4-s − 25·5-s − 144.·6-s + 29.5·7-s − 196.·8-s + 654.·9-s − 120.·10-s − 121·11-s + 260.·12-s + 345.·13-s + 142.·14-s + 749.·15-s − 670.·16-s + 2.12e3·17-s + 3.16e3·18-s + 361·19-s + 217.·20-s − 886.·21-s − 584.·22-s − 282.·23-s + 5.88e3·24-s + 625·25-s + 1.66e3·26-s − 1.23e4·27-s − 257.·28-s + ⋯
L(s)  = 1  + 0.853·2-s − 1.92·3-s − 0.271·4-s − 0.447·5-s − 1.64·6-s + 0.228·7-s − 1.08·8-s + 2.69·9-s − 0.381·10-s − 0.301·11-s + 0.521·12-s + 0.567·13-s + 0.194·14-s + 0.859·15-s − 0.654·16-s + 1.78·17-s + 2.30·18-s + 0.229·19-s + 0.121·20-s − 0.438·21-s − 0.257·22-s − 0.111·23-s + 2.08·24-s + 0.200·25-s + 0.483·26-s − 3.25·27-s − 0.0619·28-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: \(0\)
Selberg data: \((2,\ 1045,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9047654313\)
\(L(\frac12)\) \(\approx\) \(0.9047654313\)
\(L(\frac{7}{2})\) 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 \( 1 - 4.82T + 32T^{2} \)
3 \( 1 + 29.9T + 243T^{2} \)
7 \( 1 - 29.5T + 1.68e4T^{2} \)
13 \( 1 - 345.T + 3.71e5T^{2} \)
17 \( 1 - 2.12e3T + 1.41e6T^{2} \)
23 \( 1 + 282.T + 6.43e6T^{2} \)
29 \( 1 + 6.15e3T + 2.05e7T^{2} \)
31 \( 1 + 3.15e3T + 2.86e7T^{2} \)
37 \( 1 - 6.95e3T + 6.93e7T^{2} \)
41 \( 1 + 1.72e4T + 1.15e8T^{2} \)
43 \( 1 + 5.46e3T + 1.47e8T^{2} \)
47 \( 1 + 1.80e3T + 2.29e8T^{2} \)
53 \( 1 - 1.19e4T + 4.18e8T^{2} \)
59 \( 1 - 4.16e4T + 7.14e8T^{2} \)
61 \( 1 + 2.24e4T + 8.44e8T^{2} \)
67 \( 1 - 2.93e4T + 1.35e9T^{2} \)
71 \( 1 + 6.11e4T + 1.80e9T^{2} \)
73 \( 1 + 5.35e4T + 2.07e9T^{2} \)
79 \( 1 - 6.56e4T + 3.07e9T^{2} \)
83 \( 1 - 5.44e4T + 3.93e9T^{2} \)
89 \( 1 - 3.95e4T + 5.58e9T^{2} \)
97 \( 1 + 1.57e5T + 8.58e9T^{2} \)
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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

−9.487587543385505753538619119191, −8.138282446790885004450290591184, −7.24778088279792097866626758463, −6.27800339123389052802803533361, −5.47586526092051724192352789204, −5.17200357148613262952849014150, −4.13424729469429448130713915921, −3.41430321518355396143466980880, −1.44377982265690966157798419255, −0.43312702399212309942343900798, 0.43312702399212309942343900798, 1.44377982265690966157798419255, 3.41430321518355396143466980880, 4.13424729469429448130713915921, 5.17200357148613262952849014150, 5.47586526092051724192352789204, 6.27800339123389052802803533361, 7.24778088279792097866626758463, 8.138282446790885004450290591184, 9.487587543385505753538619119191

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