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  − 5.05·2-s + 9.36·3-s + 17.5·4-s + 14.1·5-s − 47.3·6-s − 13.7·7-s − 48.1·8-s + 60.6·9-s − 71.4·10-s + 10.3·11-s + 164.·12-s − 61.4·13-s + 69.2·14-s + 132.·15-s + 103.·16-s − 24.4·17-s − 306.·18-s − 65.5·19-s + 247.·20-s − 128.·21-s − 52.3·22-s + 23.4·23-s − 450.·24-s + 75.0·25-s + 310.·26-s + 315.·27-s − 240.·28-s + ⋯
L(s)  = 1  − 1.78·2-s + 1.80·3-s + 2.19·4-s + 1.26·5-s − 3.21·6-s − 0.740·7-s − 2.12·8-s + 2.24·9-s − 2.25·10-s + 0.284·11-s + 3.94·12-s − 1.31·13-s + 1.32·14-s + 2.27·15-s + 1.60·16-s − 0.348·17-s − 4.01·18-s − 0.791·19-s + 2.77·20-s − 1.33·21-s − 0.507·22-s + 0.212·23-s − 3.83·24-s + 0.600·25-s + 2.34·26-s + 2.24·27-s − 1.62·28-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 \( 1 + 5.05T + 8T^{2} \)
3 \( 1 - 9.36T + 27T^{2} \)
5 \( 1 - 14.1T + 125T^{2} \)
7 \( 1 + 13.7T + 343T^{2} \)
11 \( 1 - 10.3T + 1.33e3T^{2} \)
13 \( 1 + 61.4T + 2.19e3T^{2} \)
17 \( 1 + 24.4T + 4.91e3T^{2} \)
19 \( 1 + 65.5T + 6.85e3T^{2} \)
23 \( 1 - 23.4T + 1.21e4T^{2} \)
29 \( 1 + 236.T + 2.43e4T^{2} \)
31 \( 1 + 210.T + 2.97e4T^{2} \)
37 \( 1 + 316.T + 5.06e4T^{2} \)
41 \( 1 - 13.2T + 6.89e4T^{2} \)
47 \( 1 - 450.T + 1.03e5T^{2} \)
53 \( 1 + 205.T + 1.48e5T^{2} \)
59 \( 1 - 334.T + 2.05e5T^{2} \)
61 \( 1 - 811.T + 2.26e5T^{2} \)
67 \( 1 + 113.T + 3.00e5T^{2} \)
71 \( 1 + 53.8T + 3.57e5T^{2} \)
73 \( 1 + 108.T + 3.89e5T^{2} \)
79 \( 1 + 723.T + 4.93e5T^{2} \)
83 \( 1 - 191.T + 5.71e5T^{2} \)
89 \( 1 - 419.T + 7.04e5T^{2} \)
97 \( 1 + 953.T + 9.12e5T^{2} \)
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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

−8.823020696076416002752813940442, −7.967888083567934513246070749326, −7.09776862041647134386547774596, −6.78688484256071865756893794721, −5.52348948101699677993220916842, −3.90840441653775729035977893691, −2.77832758345628207658226916188, −2.15947265209687762044911218703, −1.60831581139716318848162360243, 0, 1.60831581139716318848162360243, 2.15947265209687762044911218703, 2.77832758345628207658226916188, 3.90840441653775729035977893691, 5.52348948101699677993220916842, 6.78688484256071865756893794721, 7.09776862041647134386547774596, 7.967888083567934513246070749326, 8.823020696076416002752813940442

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