Properties

Label 2-1045-1.1-c3-0-88
Degree $2$
Conductor $1045$
Sign $-1$
Analytic cond. $61.6569$
Root an. cond. $7.85219$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s − 3-s + 17·4-s − 5·5-s + 5·6-s − 2·7-s − 45·8-s − 26·9-s + 25·10-s − 11·11-s − 17·12-s − 7·13-s + 10·14-s + 5·15-s + 89·16-s + 14·17-s + 130·18-s + 19·19-s − 85·20-s + 2·21-s + 55·22-s + 55·23-s + 45·24-s + 25·25-s + 35·26-s + 53·27-s − 34·28-s + ⋯
L(s)  = 1  − 1.76·2-s − 0.192·3-s + 17/8·4-s − 0.447·5-s + 0.340·6-s − 0.107·7-s − 1.98·8-s − 0.962·9-s + 0.790·10-s − 0.301·11-s − 0.408·12-s − 0.149·13-s + 0.190·14-s + 0.0860·15-s + 1.39·16-s + 0.199·17-s + 1.70·18-s + 0.229·19-s − 0.950·20-s + 0.0207·21-s + 0.533·22-s + 0.498·23-s + 0.382·24-s + 1/5·25-s + 0.264·26-s + 0.377·27-s − 0.229·28-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: yes
Arithmetic: yes
Character: Trivial
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 \( 1 + p T \)
11 \( 1 + p T \)
19 \( 1 - p T \)
good2 \( 1 + 5 T + p^{3} T^{2} \)
3 \( 1 + T + p^{3} T^{2} \)
7 \( 1 + 2 T + p^{3} T^{2} \)
13 \( 1 + 7 T + p^{3} T^{2} \)
17 \( 1 - 14 T + p^{3} T^{2} \)
23 \( 1 - 55 T + p^{3} T^{2} \)
29 \( 1 + 26 T + p^{3} T^{2} \)
31 \( 1 - 261 T + p^{3} T^{2} \)
37 \( 1 + 126 T + p^{3} T^{2} \)
41 \( 1 + 381 T + p^{3} T^{2} \)
43 \( 1 - 9 p T + p^{3} T^{2} \)
47 \( 1 - 189 T + p^{3} T^{2} \)
53 \( 1 + 404 T + p^{3} T^{2} \)
59 \( 1 - 746 T + p^{3} T^{2} \)
61 \( 1 - 79 T + p^{3} T^{2} \)
67 \( 1 - 537 T + p^{3} T^{2} \)
71 \( 1 + 824 T + p^{3} T^{2} \)
73 \( 1 - 169 T + p^{3} T^{2} \)
79 \( 1 + 338 T + p^{3} T^{2} \)
83 \( 1 - 601 T + p^{3} T^{2} \)
89 \( 1 + 762 T + p^{3} T^{2} \)
97 \( 1 - 866 T + p^{3} T^{2} \)
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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

−8.993096052710292407948579233282, −8.411222985125276817454309182351, −7.72676633805505074740129848908, −6.90142010853648654468466548272, −6.05385760242674664820459871058, −4.92330676336556911263205268396, −3.29836247110109601722925020108, −2.36635898609352705579425547933, −0.969543441903787488862079112677, 0, 0.969543441903787488862079112677, 2.36635898609352705579425547933, 3.29836247110109601722925020108, 4.92330676336556911263205268396, 6.05385760242674664820459871058, 6.90142010853648654468466548272, 7.72676633805505074740129848908, 8.411222985125276817454309182351, 8.993096052710292407948579233282

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