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  + 5.20·2-s + 5.00·3-s + 19.1·4-s − 13.8·5-s + 26.0·6-s − 26.6·7-s + 57.9·8-s − 1.91·9-s − 72.1·10-s + 47.0·11-s + 95.7·12-s − 60.3·13-s − 138.·14-s − 69.3·15-s + 148.·16-s + 69.2·17-s − 9.97·18-s − 9.96·19-s − 264.·20-s − 133.·21-s + 245.·22-s − 26.9·23-s + 290.·24-s + 66.8·25-s − 314.·26-s − 144.·27-s − 509.·28-s + ⋯
L(s)  = 1  + 1.84·2-s + 0.963·3-s + 2.39·4-s − 1.23·5-s + 1.77·6-s − 1.43·7-s + 2.56·8-s − 0.0709·9-s − 2.28·10-s + 1.29·11-s + 2.30·12-s − 1.28·13-s − 2.64·14-s − 1.19·15-s + 2.32·16-s + 0.987·17-s − 0.130·18-s − 0.120·19-s − 2.96·20-s − 1.38·21-s + 2.37·22-s − 0.244·23-s + 2.46·24-s + 0.535·25-s − 2.36·26-s − 1.03·27-s − 3.43·28-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 \( 1 - 5.20T + 8T^{2} \)
3 \( 1 - 5.00T + 27T^{2} \)
5 \( 1 + 13.8T + 125T^{2} \)
7 \( 1 + 26.6T + 343T^{2} \)
11 \( 1 - 47.0T + 1.33e3T^{2} \)
13 \( 1 + 60.3T + 2.19e3T^{2} \)
17 \( 1 - 69.2T + 4.91e3T^{2} \)
19 \( 1 + 9.96T + 6.85e3T^{2} \)
23 \( 1 + 26.9T + 1.21e4T^{2} \)
29 \( 1 + 158.T + 2.43e4T^{2} \)
31 \( 1 + 129.T + 2.97e4T^{2} \)
37 \( 1 + 59.9T + 5.06e4T^{2} \)
41 \( 1 + 415.T + 6.89e4T^{2} \)
47 \( 1 + 398.T + 1.03e5T^{2} \)
53 \( 1 + 280.T + 1.48e5T^{2} \)
59 \( 1 - 143.T + 2.05e5T^{2} \)
61 \( 1 - 791.T + 2.26e5T^{2} \)
67 \( 1 + 635.T + 3.00e5T^{2} \)
71 \( 1 - 707.T + 3.57e5T^{2} \)
73 \( 1 + 168.T + 3.89e5T^{2} \)
79 \( 1 + 631.T + 4.93e5T^{2} \)
83 \( 1 - 516.T + 5.71e5T^{2} \)
89 \( 1 - 1.27e3T + 7.04e5T^{2} \)
97 \( 1 + 298.T + 9.12e5T^{2} \)
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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

−8.244642542738276137129481196263, −7.36026928524011430359894264281, −6.89294530652506111757124583227, −6.00904045272719781036612046913, −5.05288761331371495313334570515, −3.94648214645297494309903311685, −3.53027917066878753902251505247, −3.06547808369022431001092784377, −1.94653160462730410665401358324, 0, 1.94653160462730410665401358324, 3.06547808369022431001092784377, 3.53027917066878753902251505247, 3.94648214645297494309903311685, 5.05288761331371495313334570515, 6.00904045272719781036612046913, 6.89294530652506111757124583227, 7.36026928524011430359894264281, 8.244642542738276137129481196263

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