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  + 2.12·2-s − 9.87·3-s − 3.49·4-s + 15.5·5-s − 20.9·6-s + 16.3·7-s − 24.3·8-s + 70.4·9-s + 33.0·10-s + 42.7·11-s + 34.5·12-s − 44.4·13-s + 34.7·14-s − 153.·15-s − 23.8·16-s − 80.6·17-s + 149.·18-s − 9.72·19-s − 54.4·20-s − 161.·21-s + 90.6·22-s + 81.9·23-s + 240.·24-s + 117.·25-s − 94.2·26-s − 428.·27-s − 57.2·28-s + ⋯
L(s)  = 1  + 0.750·2-s − 1.89·3-s − 0.436·4-s + 1.39·5-s − 1.42·6-s + 0.884·7-s − 1.07·8-s + 2.60·9-s + 1.04·10-s + 1.17·11-s + 0.829·12-s − 0.947·13-s + 0.663·14-s − 2.64·15-s − 0.372·16-s − 1.15·17-s + 1.95·18-s − 0.117·19-s − 0.608·20-s − 1.68·21-s + 0.878·22-s + 0.743·23-s + 2.04·24-s + 0.940·25-s − 0.711·26-s − 3.05·27-s − 0.386·28-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 \( 1 - 2.12T + 8T^{2} \)
3 \( 1 + 9.87T + 27T^{2} \)
5 \( 1 - 15.5T + 125T^{2} \)
7 \( 1 - 16.3T + 343T^{2} \)
11 \( 1 - 42.7T + 1.33e3T^{2} \)
13 \( 1 + 44.4T + 2.19e3T^{2} \)
17 \( 1 + 80.6T + 4.91e3T^{2} \)
19 \( 1 + 9.72T + 6.85e3T^{2} \)
23 \( 1 - 81.9T + 1.21e4T^{2} \)
29 \( 1 + 170.T + 2.43e4T^{2} \)
31 \( 1 + 115.T + 2.97e4T^{2} \)
37 \( 1 + 56.3T + 5.06e4T^{2} \)
41 \( 1 - 9.51T + 6.89e4T^{2} \)
47 \( 1 - 9.55T + 1.03e5T^{2} \)
53 \( 1 - 254.T + 1.48e5T^{2} \)
59 \( 1 - 288.T + 2.05e5T^{2} \)
61 \( 1 - 67.9T + 2.26e5T^{2} \)
67 \( 1 - 273.T + 3.00e5T^{2} \)
71 \( 1 - 113.T + 3.57e5T^{2} \)
73 \( 1 - 908.T + 3.89e5T^{2} \)
79 \( 1 + 798.T + 4.93e5T^{2} \)
83 \( 1 + 1.27e3T + 5.71e5T^{2} \)
89 \( 1 + 1.39e3T + 7.04e5T^{2} \)
97 \( 1 - 253.T + 9.12e5T^{2} \)
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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

−8.754095350473309170090432805451, −7.16607294833637658251697805576, −6.58600150520556426390020498194, −5.79834257253302092431703859849, −5.28795508827066924587656550475, −4.70154880507063496858390134515, −3.97202649387311582884071744598, −2.14648230845092379937586656311, −1.23812506015909785015469151298, 0, 1.23812506015909785015469151298, 2.14648230845092379937586656311, 3.97202649387311582884071744598, 4.70154880507063496858390134515, 5.28795508827066924587656550475, 5.79834257253302092431703859849, 6.58600150520556426390020498194, 7.16607294833637658251697805576, 8.754095350473309170090432805451

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