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  − 0.213·2-s + 7.23·3-s − 7.95·4-s + 3.30·5-s − 1.54·6-s + 30.5·7-s + 3.41·8-s + 25.3·9-s − 0.706·10-s − 66.6·11-s − 57.5·12-s + 32.7·13-s − 6.53·14-s + 23.9·15-s + 62.9·16-s − 58.7·17-s − 5.42·18-s + 53.7·19-s − 26.2·20-s + 221.·21-s + 14.2·22-s − 169.·23-s + 24.6·24-s − 114.·25-s − 7.00·26-s − 11.9·27-s − 243.·28-s + ⋯
L(s)  = 1  − 0.0756·2-s + 1.39·3-s − 0.994·4-s + 0.295·5-s − 0.105·6-s + 1.65·7-s + 0.150·8-s + 0.938·9-s − 0.0223·10-s − 1.82·11-s − 1.38·12-s + 0.698·13-s − 0.124·14-s + 0.411·15-s + 0.982·16-s − 0.837·17-s − 0.0709·18-s + 0.649·19-s − 0.294·20-s + 2.29·21-s + 0.138·22-s − 1.53·23-s + 0.209·24-s − 0.912·25-s − 0.0528·26-s − 0.0849·27-s − 1.64·28-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 \( 1 + 0.213T + 8T^{2} \)
3 \( 1 - 7.23T + 27T^{2} \)
5 \( 1 - 3.30T + 125T^{2} \)
7 \( 1 - 30.5T + 343T^{2} \)
11 \( 1 + 66.6T + 1.33e3T^{2} \)
13 \( 1 - 32.7T + 2.19e3T^{2} \)
17 \( 1 + 58.7T + 4.91e3T^{2} \)
19 \( 1 - 53.7T + 6.85e3T^{2} \)
23 \( 1 + 169.T + 1.21e4T^{2} \)
29 \( 1 + 67.9T + 2.43e4T^{2} \)
31 \( 1 + 2.72T + 2.97e4T^{2} \)
37 \( 1 + 335.T + 5.06e4T^{2} \)
41 \( 1 - 113.T + 6.89e4T^{2} \)
47 \( 1 + 24.6T + 1.03e5T^{2} \)
53 \( 1 + 597.T + 1.48e5T^{2} \)
59 \( 1 - 205.T + 2.05e5T^{2} \)
61 \( 1 + 392.T + 2.26e5T^{2} \)
67 \( 1 - 607.T + 3.00e5T^{2} \)
71 \( 1 - 98.1T + 3.57e5T^{2} \)
73 \( 1 + 874.T + 3.89e5T^{2} \)
79 \( 1 - 788.T + 4.93e5T^{2} \)
83 \( 1 + 495.T + 5.71e5T^{2} \)
89 \( 1 - 528.T + 7.04e5T^{2} \)
97 \( 1 - 2.16T + 9.12e5T^{2} \)
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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

−8.341005517370873595726996050605, −8.008016446225899963198630793391, −7.51024253202192400671059025965, −5.79881176367497587685502491560, −5.11365212023176254933923598621, −4.31042036550959526311334692016, −3.44448883571955601531361825942, −2.27789840407385858825632378030, −1.61961822146114608508061610694, 0, 1.61961822146114608508061610694, 2.27789840407385858825632378030, 3.44448883571955601531361825942, 4.31042036550959526311334692016, 5.11365212023176254933923598621, 5.79881176367497587685502491560, 7.51024253202192400671059025965, 8.008016446225899963198630793391, 8.341005517370873595726996050605

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