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  + 1.05·2-s − 4.53·3-s − 6.88·4-s − 14.4·5-s − 4.78·6-s − 15.4·7-s − 15.6·8-s − 6.42·9-s − 15.2·10-s − 62.4·11-s + 31.2·12-s − 21.6·13-s − 16.3·14-s + 65.6·15-s + 38.5·16-s + 25.1·17-s − 6.77·18-s + 45.2·19-s + 99.6·20-s + 70.1·21-s − 65.8·22-s + 195.·23-s + 71.2·24-s + 84.4·25-s − 22.7·26-s + 151.·27-s + 106.·28-s + ⋯
L(s)  = 1  + 0.372·2-s − 0.872·3-s − 0.861·4-s − 1.29·5-s − 0.325·6-s − 0.835·7-s − 0.693·8-s − 0.237·9-s − 0.482·10-s − 1.71·11-s + 0.751·12-s − 0.461·13-s − 0.311·14-s + 1.13·15-s + 0.602·16-s + 0.358·17-s − 0.0886·18-s + 0.546·19-s + 1.11·20-s + 0.729·21-s − 0.638·22-s + 1.76·23-s + 0.605·24-s + 0.675·25-s − 0.171·26-s + 1.08·27-s + 0.719·28-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 \( 1 - 1.05T + 8T^{2} \)
3 \( 1 + 4.53T + 27T^{2} \)
5 \( 1 + 14.4T + 125T^{2} \)
7 \( 1 + 15.4T + 343T^{2} \)
11 \( 1 + 62.4T + 1.33e3T^{2} \)
13 \( 1 + 21.6T + 2.19e3T^{2} \)
17 \( 1 - 25.1T + 4.91e3T^{2} \)
19 \( 1 - 45.2T + 6.85e3T^{2} \)
23 \( 1 - 195.T + 1.21e4T^{2} \)
29 \( 1 + 39.6T + 2.43e4T^{2} \)
31 \( 1 + 254.T + 2.97e4T^{2} \)
37 \( 1 + 110.T + 5.06e4T^{2} \)
41 \( 1 + 62.1T + 6.89e4T^{2} \)
47 \( 1 - 586.T + 1.03e5T^{2} \)
53 \( 1 + 515.T + 1.48e5T^{2} \)
59 \( 1 - 250.T + 2.05e5T^{2} \)
61 \( 1 - 812.T + 2.26e5T^{2} \)
67 \( 1 + 558.T + 3.00e5T^{2} \)
71 \( 1 - 227.T + 3.57e5T^{2} \)
73 \( 1 + 568.T + 3.89e5T^{2} \)
79 \( 1 + 1.33e3T + 4.93e5T^{2} \)
83 \( 1 + 16.2T + 5.71e5T^{2} \)
89 \( 1 - 858.T + 7.04e5T^{2} \)
97 \( 1 + 312.T + 9.12e5T^{2} \)
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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

−8.474783620498820384799845049241, −7.60021452049986626649225948709, −7.00349411400900112997897044229, −5.67701264576644834715819454933, −5.29942100283029719334283825407, −4.53113425308273676054294331914, −3.42356732966095042715548034622, −2.89531892146267553482947843292, −0.61406931325781491862928900344, 0, 0.61406931325781491862928900344, 2.89531892146267553482947843292, 3.42356732966095042715548034622, 4.53113425308273676054294331914, 5.29942100283029719334283825407, 5.67701264576644834715819454933, 7.00349411400900112997897044229, 7.60021452049986626649225948709, 8.474783620498820384799845049241

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