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  + 3.07·2-s + 5.77·3-s + 1.48·4-s + 13.4·5-s + 17.7·6-s − 34.2·7-s − 20.0·8-s + 6.35·9-s + 41.4·10-s + 24.2·11-s + 8.55·12-s + 11.8·13-s − 105.·14-s + 77.8·15-s − 73.6·16-s + 110.·17-s + 19.5·18-s − 45.6·19-s + 19.9·20-s − 197.·21-s + 74.8·22-s − 214.·23-s − 115.·24-s + 56.5·25-s + 36.3·26-s − 119.·27-s − 50.7·28-s + ⋯
L(s)  = 1  + 1.08·2-s + 1.11·3-s + 0.185·4-s + 1.20·5-s + 1.20·6-s − 1.84·7-s − 0.887·8-s + 0.235·9-s + 1.31·10-s + 0.665·11-s + 0.205·12-s + 0.252·13-s − 2.01·14-s + 1.33·15-s − 1.15·16-s + 1.58·17-s + 0.256·18-s − 0.551·19-s + 0.223·20-s − 2.05·21-s + 0.724·22-s − 1.94·23-s − 0.985·24-s + 0.452·25-s + 0.274·26-s − 0.849·27-s − 0.342·28-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 \( 1 - 3.07T + 8T^{2} \)
3 \( 1 - 5.77T + 27T^{2} \)
5 \( 1 - 13.4T + 125T^{2} \)
7 \( 1 + 34.2T + 343T^{2} \)
11 \( 1 - 24.2T + 1.33e3T^{2} \)
13 \( 1 - 11.8T + 2.19e3T^{2} \)
17 \( 1 - 110.T + 4.91e3T^{2} \)
19 \( 1 + 45.6T + 6.85e3T^{2} \)
23 \( 1 + 214.T + 1.21e4T^{2} \)
29 \( 1 - 57.4T + 2.43e4T^{2} \)
31 \( 1 + 181.T + 2.97e4T^{2} \)
37 \( 1 + 156.T + 5.06e4T^{2} \)
41 \( 1 - 0.775T + 6.89e4T^{2} \)
47 \( 1 + 146.T + 1.03e5T^{2} \)
53 \( 1 + 282.T + 1.48e5T^{2} \)
59 \( 1 + 698.T + 2.05e5T^{2} \)
61 \( 1 + 98.7T + 2.26e5T^{2} \)
67 \( 1 + 517.T + 3.00e5T^{2} \)
71 \( 1 - 29.8T + 3.57e5T^{2} \)
73 \( 1 + 75.4T + 3.89e5T^{2} \)
79 \( 1 - 882.T + 4.93e5T^{2} \)
83 \( 1 - 716.T + 5.71e5T^{2} \)
89 \( 1 - 142.T + 7.04e5T^{2} \)
97 \( 1 + 352.T + 9.12e5T^{2} \)
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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

−8.735457714736029219786493725532, −7.68862719062665194482381837981, −6.38935038881063016620077753388, −6.13925059794967472644081738255, −5.40531507370851435133174313746, −3.95307734801759411056186416329, −3.46605369207622023238838256486, −2.78934564583340686575873725804, −1.79671012116075412216482574670, 0, 1.79671012116075412216482574670, 2.78934564583340686575873725804, 3.46605369207622023238838256486, 3.95307734801759411056186416329, 5.40531507370851435133174313746, 6.13925059794967472644081738255, 6.38935038881063016620077753388, 7.68862719062665194482381837981, 8.735457714736029219786493725532

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