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  − 3.81·2-s + 8.34·3-s + 6.58·4-s + 2.56·5-s − 31.8·6-s − 3.42·7-s + 5.39·8-s + 42.6·9-s − 9.79·10-s − 24.5·11-s + 54.9·12-s − 46.2·13-s + 13.0·14-s + 21.4·15-s − 73.3·16-s + 97.5·17-s − 163.·18-s − 131.·19-s + 16.8·20-s − 28.5·21-s + 93.6·22-s + 175.·23-s + 45.0·24-s − 118.·25-s + 176.·26-s + 130.·27-s − 22.5·28-s + ⋯
L(s)  = 1  − 1.35·2-s + 1.60·3-s + 0.823·4-s + 0.229·5-s − 2.16·6-s − 0.184·7-s + 0.238·8-s + 1.58·9-s − 0.309·10-s − 0.672·11-s + 1.32·12-s − 0.987·13-s + 0.249·14-s + 0.368·15-s − 1.14·16-s + 1.39·17-s − 2.13·18-s − 1.59·19-s + 0.188·20-s − 0.296·21-s + 0.907·22-s + 1.58·23-s + 0.383·24-s − 0.947·25-s + 1.33·26-s + 0.933·27-s − 0.152·28-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 \( 1 + 3.81T + 8T^{2} \)
3 \( 1 - 8.34T + 27T^{2} \)
5 \( 1 - 2.56T + 125T^{2} \)
7 \( 1 + 3.42T + 343T^{2} \)
11 \( 1 + 24.5T + 1.33e3T^{2} \)
13 \( 1 + 46.2T + 2.19e3T^{2} \)
17 \( 1 - 97.5T + 4.91e3T^{2} \)
19 \( 1 + 131.T + 6.85e3T^{2} \)
23 \( 1 - 175.T + 1.21e4T^{2} \)
29 \( 1 + 83.1T + 2.43e4T^{2} \)
31 \( 1 - 128.T + 2.97e4T^{2} \)
37 \( 1 - 301.T + 5.06e4T^{2} \)
41 \( 1 + 201.T + 6.89e4T^{2} \)
47 \( 1 + 335.T + 1.03e5T^{2} \)
53 \( 1 - 373.T + 1.48e5T^{2} \)
59 \( 1 - 578.T + 2.05e5T^{2} \)
61 \( 1 + 892.T + 2.26e5T^{2} \)
67 \( 1 - 2.66T + 3.00e5T^{2} \)
71 \( 1 - 467.T + 3.57e5T^{2} \)
73 \( 1 + 481.T + 3.89e5T^{2} \)
79 \( 1 + 296.T + 4.93e5T^{2} \)
83 \( 1 - 122.T + 5.71e5T^{2} \)
89 \( 1 - 746.T + 7.04e5T^{2} \)
97 \( 1 + 1.72e3T + 9.12e5T^{2} \)
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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

−8.477341974157166994639654911350, −7.943606385962808013647024758444, −7.44994089989908877556145592511, −6.56203972606327802059043692207, −5.16403590861802549616887963501, −4.14521042248147534634623611255, −2.96597054208219591772978239788, −2.32157548281087653833785398468, −1.36499101276782826890288764270, 0, 1.36499101276782826890288764270, 2.32157548281087653833785398468, 2.96597054208219591772978239788, 4.14521042248147534634623611255, 5.16403590861802549616887963501, 6.56203972606327802059043692207, 7.44994089989908877556145592511, 7.943606385962808013647024758444, 8.477341974157166994639654911350

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