Properties

Label 2-1045-1.1-c3-0-132
Degree $2$
Conductor $1045$
Sign $-1$
Analytic cond. $61.6569$
Root an. cond. $7.85219$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.81·2-s + 3.33·3-s − 0.0697·4-s − 5·5-s − 9.39·6-s + 35.3·7-s + 22.7·8-s − 15.8·9-s + 14.0·10-s − 11·11-s − 0.232·12-s + 16.5·13-s − 99.6·14-s − 16.6·15-s − 63.4·16-s − 35.7·17-s + 44.6·18-s + 19·19-s + 0.348·20-s + 118.·21-s + 30.9·22-s − 96.3·23-s + 75.8·24-s + 25·25-s − 46.6·26-s − 143.·27-s − 2.46·28-s + ⋯
L(s)  = 1  − 0.995·2-s + 0.642·3-s − 0.00871·4-s − 0.447·5-s − 0.639·6-s + 1.91·7-s + 1.00·8-s − 0.587·9-s + 0.445·10-s − 0.301·11-s − 0.00559·12-s + 0.353·13-s − 1.90·14-s − 0.287·15-s − 0.991·16-s − 0.509·17-s + 0.585·18-s + 0.229·19-s + 0.00389·20-s + 1.22·21-s + 0.300·22-s − 0.873·23-s + 0.644·24-s + 0.200·25-s − 0.351·26-s − 1.01·27-s − 0.0166·28-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: no
Arithmetic: yes
Character: $\chi_{1045} (1, \cdot )$
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 \( 1 + 5T \)
11 \( 1 + 11T \)
19 \( 1 - 19T \)
good2 \( 1 + 2.81T + 8T^{2} \)
3 \( 1 - 3.33T + 27T^{2} \)
7 \( 1 - 35.3T + 343T^{2} \)
13 \( 1 - 16.5T + 2.19e3T^{2} \)
17 \( 1 + 35.7T + 4.91e3T^{2} \)
23 \( 1 + 96.3T + 1.21e4T^{2} \)
29 \( 1 + 163.T + 2.43e4T^{2} \)
31 \( 1 - 130.T + 2.97e4T^{2} \)
37 \( 1 - 7.26T + 5.06e4T^{2} \)
41 \( 1 + 315.T + 6.89e4T^{2} \)
43 \( 1 + 250.T + 7.95e4T^{2} \)
47 \( 1 + 280.T + 1.03e5T^{2} \)
53 \( 1 - 504.T + 1.48e5T^{2} \)
59 \( 1 - 49.5T + 2.05e5T^{2} \)
61 \( 1 - 360.T + 2.26e5T^{2} \)
67 \( 1 + 370.T + 3.00e5T^{2} \)
71 \( 1 - 309.T + 3.57e5T^{2} \)
73 \( 1 - 42.6T + 3.89e5T^{2} \)
79 \( 1 + 771.T + 4.93e5T^{2} \)
83 \( 1 + 363.T + 5.71e5T^{2} \)
89 \( 1 - 0.664T + 7.04e5T^{2} \)
97 \( 1 + 573.T + 9.12e5T^{2} \)
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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

−8.773868948064443065649660419730, −8.324536946128196805794703803554, −7.955259124614380181059579893441, −7.09942321473050177309614061847, −5.53193968267876566002908428307, −4.69094490856138721717637199674, −3.78456038516061115208535469764, −2.27244047346374304980837966031, −1.39690618399071779834530675370, 0, 1.39690618399071779834530675370, 2.27244047346374304980837966031, 3.78456038516061115208535469764, 4.69094490856138721717637199674, 5.53193968267876566002908428307, 7.09942321473050177309614061847, 7.955259124614380181059579893441, 8.324536946128196805794703803554, 8.773868948064443065649660419730

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