Properties

Degree $2$
Conductor $1849$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.53·2-s + 8.01·3-s − 1.58·4-s + 20.7·5-s − 20.2·6-s + 3.06·7-s + 24.2·8-s + 37.2·9-s − 52.4·10-s − 50.3·11-s − 12.7·12-s − 26.3·13-s − 7.75·14-s + 165.·15-s − 48.7·16-s − 23.1·17-s − 94.2·18-s − 135.·19-s − 32.8·20-s + 24.5·21-s + 127.·22-s − 91.8·23-s + 194.·24-s + 304.·25-s + 66.7·26-s + 81.8·27-s − 4.85·28-s + ⋯
L(s)  = 1  − 0.895·2-s + 1.54·3-s − 0.198·4-s + 1.85·5-s − 1.38·6-s + 0.165·7-s + 1.07·8-s + 1.37·9-s − 1.65·10-s − 1.37·11-s − 0.305·12-s − 0.562·13-s − 0.147·14-s + 2.85·15-s − 0.762·16-s − 0.329·17-s − 1.23·18-s − 1.64·19-s − 0.367·20-s + 0.254·21-s + 1.23·22-s − 0.832·23-s + 1.65·24-s + 2.43·25-s + 0.503·26-s + 0.583·27-s − 0.0327·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-1$
Motivic weight: \(3\)
Character: $\chi_{1849} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1849,\ (\ :3/2),\ -1)\)

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)$
bad43 \( 1 \)
good2 \( 1 + 2.53T + 8T^{2} \)
3 \( 1 - 8.01T + 27T^{2} \)
5 \( 1 - 20.7T + 125T^{2} \)
7 \( 1 - 3.06T + 343T^{2} \)
11 \( 1 + 50.3T + 1.33e3T^{2} \)
13 \( 1 + 26.3T + 2.19e3T^{2} \)
17 \( 1 + 23.1T + 4.91e3T^{2} \)
19 \( 1 + 135.T + 6.85e3T^{2} \)
23 \( 1 + 91.8T + 1.21e4T^{2} \)
29 \( 1 + 15.5T + 2.43e4T^{2} \)
31 \( 1 + 34.9T + 2.97e4T^{2} \)
37 \( 1 + 221.T + 5.06e4T^{2} \)
41 \( 1 - 206.T + 6.89e4T^{2} \)
47 \( 1 + 600.T + 1.03e5T^{2} \)
53 \( 1 - 408.T + 1.48e5T^{2} \)
59 \( 1 + 740.T + 2.05e5T^{2} \)
61 \( 1 - 51.7T + 2.26e5T^{2} \)
67 \( 1 - 334.T + 3.00e5T^{2} \)
71 \( 1 + 487.T + 3.57e5T^{2} \)
73 \( 1 + 722.T + 3.89e5T^{2} \)
79 \( 1 + 171.T + 4.93e5T^{2} \)
83 \( 1 + 241.T + 5.71e5T^{2} \)
89 \( 1 + 333.T + 7.04e5T^{2} \)
97 \( 1 - 687.T + 9.12e5T^{2} \)
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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

−8.652316226089819009995116837714, −8.054330951049633346492337600158, −7.27030402356795367462713199485, −6.22432453568851754391108688798, −5.17565455757664358090681596148, −4.38483972406086336913544154334, −2.95814723554666080438807070795, −2.09442896443710035624456607069, −1.75306264457445927967283020093, 0, 1.75306264457445927967283020093, 2.09442896443710035624456607069, 2.95814723554666080438807070795, 4.38483972406086336913544154334, 5.17565455757664358090681596148, 6.22432453568851754391108688798, 7.27030402356795367462713199485, 8.054330951049633346492337600158, 8.652316226089819009995116837714

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