Properties

Label 2-1045-1.1-c5-0-50
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  − 9.16·2-s − 28.2·3-s + 52.0·4-s + 25·5-s + 259.·6-s + 66.4·7-s − 183.·8-s + 555.·9-s − 229.·10-s − 121·11-s − 1.47e3·12-s + 856.·13-s − 608.·14-s − 706.·15-s + 18.3·16-s − 2.25e3·17-s − 5.09e3·18-s − 361·19-s + 1.30e3·20-s − 1.87e3·21-s + 1.10e3·22-s + 2.09e3·23-s + 5.18e3·24-s + 625·25-s − 7.84e3·26-s − 8.82e3·27-s + 3.45e3·28-s + ⋯
L(s)  = 1  − 1.62·2-s − 1.81·3-s + 1.62·4-s + 0.447·5-s + 2.93·6-s + 0.512·7-s − 1.01·8-s + 2.28·9-s − 0.724·10-s − 0.301·11-s − 2.94·12-s + 1.40·13-s − 0.830·14-s − 0.810·15-s + 0.0178·16-s − 1.88·17-s − 3.70·18-s − 0.229·19-s + 0.727·20-s − 0.928·21-s + 0.488·22-s + 0.825·23-s + 1.83·24-s + 0.200·25-s − 2.27·26-s − 2.33·27-s + 0.833·28-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

\(L(3)\) \(\approx\) \(0.4298225190\)
\(L(\frac12)\) \(\approx\) \(0.4298225190\)
\(L(\frac{7}{2})\) 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 \( 1 + 9.16T + 32T^{2} \)
3 \( 1 + 28.2T + 243T^{2} \)
7 \( 1 - 66.4T + 1.68e4T^{2} \)
13 \( 1 - 856.T + 3.71e5T^{2} \)
17 \( 1 + 2.25e3T + 1.41e6T^{2} \)
23 \( 1 - 2.09e3T + 6.43e6T^{2} \)
29 \( 1 + 3.55e3T + 2.05e7T^{2} \)
31 \( 1 + 6.11e3T + 2.86e7T^{2} \)
37 \( 1 - 1.18e4T + 6.93e7T^{2} \)
41 \( 1 + 4.11e3T + 1.15e8T^{2} \)
43 \( 1 + 1.51e4T + 1.47e8T^{2} \)
47 \( 1 - 2.92e4T + 2.29e8T^{2} \)
53 \( 1 - 8.10e3T + 4.18e8T^{2} \)
59 \( 1 - 3.53e4T + 7.14e8T^{2} \)
61 \( 1 - 1.37e4T + 8.44e8T^{2} \)
67 \( 1 + 2.35e4T + 1.35e9T^{2} \)
71 \( 1 - 2.29e4T + 1.80e9T^{2} \)
73 \( 1 + 5.33e4T + 2.07e9T^{2} \)
79 \( 1 + 3.79e4T + 3.07e9T^{2} \)
83 \( 1 - 6.53e4T + 3.93e9T^{2} \)
89 \( 1 - 8.13e4T + 5.58e9T^{2} \)
97 \( 1 - 5.79e4T + 8.58e9T^{2} \)
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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

−9.181825003363411713520470954077, −8.589507018734637298902665045808, −7.44233083772489298144000517157, −6.70779350225014415430391899849, −6.08641238381872534438277299913, −5.15596676245557534840833284720, −4.15790763372507656439039853303, −2.12737294106529977215345423472, −1.30439432735679852562697302930, −0.45040129119925312771928707913, 0.45040129119925312771928707913, 1.30439432735679852562697302930, 2.12737294106529977215345423472, 4.15790763372507656439039853303, 5.15596676245557534840833284720, 6.08641238381872534438277299913, 6.70779350225014415430391899849, 7.44233083772489298144000517157, 8.589507018734637298902665045808, 9.181825003363411713520470954077

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