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  − 4.52·2-s − 1.75·3-s + 12.4·4-s − 13.0·5-s + 7.91·6-s + 4.32·7-s − 20.0·8-s − 23.9·9-s + 58.9·10-s − 55.2·11-s − 21.7·12-s − 37.7·13-s − 19.5·14-s + 22.8·15-s − 8.85·16-s − 24.9·17-s + 108.·18-s + 39.7·19-s − 162.·20-s − 7.57·21-s + 249.·22-s − 2.55·23-s + 35.1·24-s + 45.2·25-s + 170.·26-s + 89.1·27-s + 53.8·28-s + ⋯
L(s)  = 1  − 1.59·2-s − 0.337·3-s + 1.55·4-s − 1.16·5-s + 0.538·6-s + 0.233·7-s − 0.886·8-s − 0.886·9-s + 1.86·10-s − 1.51·11-s − 0.523·12-s − 0.805·13-s − 0.373·14-s + 0.393·15-s − 0.138·16-s − 0.355·17-s + 1.41·18-s + 0.479·19-s − 1.81·20-s − 0.0787·21-s + 2.42·22-s − 0.0231·23-s + 0.298·24-s + 0.361·25-s + 1.28·26-s + 0.635·27-s + 0.363·28-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 \( 1 + 4.52T + 8T^{2} \)
3 \( 1 + 1.75T + 27T^{2} \)
5 \( 1 + 13.0T + 125T^{2} \)
7 \( 1 - 4.32T + 343T^{2} \)
11 \( 1 + 55.2T + 1.33e3T^{2} \)
13 \( 1 + 37.7T + 2.19e3T^{2} \)
17 \( 1 + 24.9T + 4.91e3T^{2} \)
19 \( 1 - 39.7T + 6.85e3T^{2} \)
23 \( 1 + 2.55T + 1.21e4T^{2} \)
29 \( 1 + 238.T + 2.43e4T^{2} \)
31 \( 1 - 263.T + 2.97e4T^{2} \)
37 \( 1 - 353.T + 5.06e4T^{2} \)
41 \( 1 - 254.T + 6.89e4T^{2} \)
47 \( 1 + 386.T + 1.03e5T^{2} \)
53 \( 1 + 35.7T + 1.48e5T^{2} \)
59 \( 1 + 87.4T + 2.05e5T^{2} \)
61 \( 1 - 890.T + 2.26e5T^{2} \)
67 \( 1 + 998.T + 3.00e5T^{2} \)
71 \( 1 - 443.T + 3.57e5T^{2} \)
73 \( 1 + 70.5T + 3.89e5T^{2} \)
79 \( 1 - 975.T + 4.93e5T^{2} \)
83 \( 1 - 836.T + 5.71e5T^{2} \)
89 \( 1 + 1.08e3T + 7.04e5T^{2} \)
97 \( 1 + 1.13e3T + 9.12e5T^{2} \)
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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

−8.247518752959554632551399326740, −7.903247342097837171953951864223, −7.43301394083697658747133741864, −6.38101686693347130259747263325, −5.29647442526563424100151693763, −4.46110722593405980695519180638, −3.02886040395183892884935578404, −2.23630854055838839734230141502, −0.67266851786011116871264386713, 0, 0.67266851786011116871264386713, 2.23630854055838839734230141502, 3.02886040395183892884935578404, 4.46110722593405980695519180638, 5.29647442526563424100151693763, 6.38101686693347130259747263325, 7.43301394083697658747133741864, 7.903247342097837171953951864223, 8.247518752959554632551399326740

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