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  − 3.43·2-s − 1.13·3-s + 3.79·4-s − 2.82·5-s + 3.90·6-s − 33.5·7-s + 14.4·8-s − 25.7·9-s + 9.71·10-s − 8.65·11-s − 4.31·12-s + 54.3·13-s + 115.·14-s + 3.21·15-s − 79.9·16-s − 76.6·17-s + 88.3·18-s + 95.8·19-s − 10.7·20-s + 38.1·21-s + 29.7·22-s − 162.·23-s − 16.4·24-s − 116.·25-s − 186.·26-s + 59.8·27-s − 127.·28-s + ⋯
L(s)  = 1  − 1.21·2-s − 0.218·3-s + 0.474·4-s − 0.252·5-s + 0.265·6-s − 1.81·7-s + 0.637·8-s − 0.952·9-s + 0.307·10-s − 0.237·11-s − 0.103·12-s + 1.15·13-s + 2.20·14-s + 0.0553·15-s − 1.24·16-s − 1.09·17-s + 1.15·18-s + 1.15·19-s − 0.120·20-s + 0.396·21-s + 0.287·22-s − 1.47·23-s − 0.139·24-s − 0.935·25-s − 1.40·26-s + 0.426·27-s − 0.860·28-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 \( 1 + 3.43T + 8T^{2} \)
3 \( 1 + 1.13T + 27T^{2} \)
5 \( 1 + 2.82T + 125T^{2} \)
7 \( 1 + 33.5T + 343T^{2} \)
11 \( 1 + 8.65T + 1.33e3T^{2} \)
13 \( 1 - 54.3T + 2.19e3T^{2} \)
17 \( 1 + 76.6T + 4.91e3T^{2} \)
19 \( 1 - 95.8T + 6.85e3T^{2} \)
23 \( 1 + 162.T + 1.21e4T^{2} \)
29 \( 1 - 17.8T + 2.43e4T^{2} \)
31 \( 1 - 135.T + 2.97e4T^{2} \)
37 \( 1 - 204.T + 5.06e4T^{2} \)
41 \( 1 + 302.T + 6.89e4T^{2} \)
47 \( 1 + 79.5T + 1.03e5T^{2} \)
53 \( 1 - 445.T + 1.48e5T^{2} \)
59 \( 1 + 219.T + 2.05e5T^{2} \)
61 \( 1 - 116.T + 2.26e5T^{2} \)
67 \( 1 - 859.T + 3.00e5T^{2} \)
71 \( 1 - 1.02e3T + 3.57e5T^{2} \)
73 \( 1 - 136.T + 3.89e5T^{2} \)
79 \( 1 + 584.T + 4.93e5T^{2} \)
83 \( 1 - 27.3T + 5.71e5T^{2} \)
89 \( 1 + 1.12e3T + 7.04e5T^{2} \)
97 \( 1 - 643.T + 9.12e5T^{2} \)
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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

−8.493775958356192120484105117086, −8.038318708237982719816657449122, −6.92088025653202566715433867197, −6.31113160777776949566561965422, −5.55737386846417148709565292843, −4.14815926552829743914178861868, −3.32719257110981792212942619580, −2.25800284717244846207870984907, −0.74271620810225188644116146328, 0, 0.74271620810225188644116146328, 2.25800284717244846207870984907, 3.32719257110981792212942619580, 4.14815926552829743914178861868, 5.55737386846417148709565292843, 6.31113160777776949566561965422, 6.92088025653202566715433867197, 8.038318708237982719816657449122, 8.493775958356192120484105117086

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