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  + 1.02·2-s − 1.11·3-s − 6.95·4-s − 17.8·5-s − 1.13·6-s − 19.7·7-s − 15.2·8-s − 25.7·9-s − 18.2·10-s + 22.7·11-s + 7.71·12-s + 19.7·13-s − 20.1·14-s + 19.7·15-s + 39.9·16-s − 101.·17-s − 26.3·18-s + 71.2·19-s + 123.·20-s + 21.9·21-s + 23.3·22-s + 39.5·23-s + 16.9·24-s + 192.·25-s + 20.2·26-s + 58.5·27-s + 137.·28-s + ⋯
L(s)  = 1  + 0.361·2-s − 0.213·3-s − 0.869·4-s − 1.59·5-s − 0.0772·6-s − 1.06·7-s − 0.675·8-s − 0.954·9-s − 0.576·10-s + 0.624·11-s + 0.185·12-s + 0.421·13-s − 0.385·14-s + 0.340·15-s + 0.624·16-s − 1.45·17-s − 0.345·18-s + 0.860·19-s + 1.38·20-s + 0.227·21-s + 0.225·22-s + 0.358·23-s + 0.144·24-s + 1.54·25-s + 0.152·26-s + 0.417·27-s + 0.926·28-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 \( 1 - 1.02T + 8T^{2} \)
3 \( 1 + 1.11T + 27T^{2} \)
5 \( 1 + 17.8T + 125T^{2} \)
7 \( 1 + 19.7T + 343T^{2} \)
11 \( 1 - 22.7T + 1.33e3T^{2} \)
13 \( 1 - 19.7T + 2.19e3T^{2} \)
17 \( 1 + 101.T + 4.91e3T^{2} \)
19 \( 1 - 71.2T + 6.85e3T^{2} \)
23 \( 1 - 39.5T + 1.21e4T^{2} \)
29 \( 1 - 100.T + 2.43e4T^{2} \)
31 \( 1 - 188.T + 2.97e4T^{2} \)
37 \( 1 - 431.T + 5.06e4T^{2} \)
41 \( 1 + 388.T + 6.89e4T^{2} \)
47 \( 1 + 444.T + 1.03e5T^{2} \)
53 \( 1 + 561.T + 1.48e5T^{2} \)
59 \( 1 + 582.T + 2.05e5T^{2} \)
61 \( 1 - 490.T + 2.26e5T^{2} \)
67 \( 1 - 638.T + 3.00e5T^{2} \)
71 \( 1 - 37.3T + 3.57e5T^{2} \)
73 \( 1 + 504.T + 3.89e5T^{2} \)
79 \( 1 - 690.T + 4.93e5T^{2} \)
83 \( 1 + 112.T + 5.71e5T^{2} \)
89 \( 1 - 772.T + 7.04e5T^{2} \)
97 \( 1 + 403.T + 9.12e5T^{2} \)
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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

−8.469985624780428215713053211406, −7.916287231030001891823479293051, −6.63801880472538996693589871060, −6.25974805708967487779435168595, −4.97055334576475624873965417739, −4.35507546024553913765716835271, −3.44677313350485098749546732297, −2.98099489657457279686552201482, −0.76586288639847085482345866285, 0, 0.76586288639847085482345866285, 2.98099489657457279686552201482, 3.44677313350485098749546732297, 4.35507546024553913765716835271, 4.97055334576475624873965417739, 6.25974805708967487779435168595, 6.63801880472538996693589871060, 7.916287231030001891823479293051, 8.469985624780428215713053211406

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