Properties

Degree $2$
Conductor $1849$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.44·2-s + 2.44·3-s + 3.99·4-s + 2.44·5-s − 5.99·6-s − 2.44·7-s − 4.89·8-s + 2.99·9-s − 5.99·10-s − 11-s + 9.79·12-s − 3·13-s + 5.99·14-s + 5.99·15-s + 3.99·16-s − 7·17-s − 7.34·18-s − 4.89·19-s + 9.79·20-s − 5.99·21-s + 2.44·22-s + 23-s − 11.9·24-s + 0.999·25-s + 7.34·26-s − 9.79·28-s + 2.44·29-s + ⋯
L(s)  = 1  − 1.73·2-s + 1.41·3-s + 1.99·4-s + 1.09·5-s − 2.44·6-s − 0.925·7-s − 1.73·8-s + 0.999·9-s − 1.89·10-s − 0.301·11-s + 2.82·12-s − 0.832·13-s + 1.60·14-s + 1.54·15-s + 0.999·16-s − 1.69·17-s − 1.73·18-s − 1.12·19-s + 2.19·20-s − 1.30·21-s + 0.522·22-s + 0.208·23-s − 2.44·24-s + 0.199·25-s + 1.44·26-s − 1.85·28-s + 0.454·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{1849} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1849,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.44T + 2T^{2} \)
3 \( 1 - 2.44T + 3T^{2} \)
5 \( 1 - 2.44T + 5T^{2} \)
7 \( 1 + 2.44T + 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 + 4.89T + 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 - 2.44T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 4.89T + 37T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
47 \( 1 + 10T + 47T^{2} \)
53 \( 1 + T + 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 + 7.34T + 61T^{2} \)
67 \( 1 - 9T + 67T^{2} \)
71 \( 1 + 4.89T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 - T + 83T^{2} \)
89 \( 1 - 17.1T + 89T^{2} \)
97 \( 1 - 11T + 97T^{2} \)
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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

−9.073381058447805601256365309005, −8.337089079911450708883195215285, −7.60584991633589573479032289791, −6.67863814172410917138263516897, −6.25002369939981202080222924059, −4.66217848583934006532747445107, −3.19392624763796702691461080328, −2.31122467508602671580007165175, −1.92387577974938266647656403220, 0, 1.92387577974938266647656403220, 2.31122467508602671580007165175, 3.19392624763796702691461080328, 4.66217848583934006532747445107, 6.25002369939981202080222924059, 6.67863814172410917138263516897, 7.60584991633589573479032289791, 8.337089079911450708883195215285, 9.073381058447805601256365309005

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