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  − 3.53·2-s + 7.81·3-s + 4.49·4-s + 1.07·5-s − 27.6·6-s − 33.3·7-s + 12.3·8-s + 34.1·9-s − 3.81·10-s − 6.35·11-s + 35.1·12-s + 16.4·13-s + 117.·14-s + 8.44·15-s − 79.7·16-s + 8.35·17-s − 120.·18-s + 141.·19-s + 4.85·20-s − 260.·21-s + 22.4·22-s + 123.·23-s + 96.8·24-s − 123.·25-s − 58.2·26-s + 55.5·27-s − 149.·28-s + ⋯
L(s)  = 1  − 1.24·2-s + 1.50·3-s + 0.561·4-s + 0.0965·5-s − 1.88·6-s − 1.79·7-s + 0.547·8-s + 1.26·9-s − 0.120·10-s − 0.174·11-s + 0.845·12-s + 0.351·13-s + 2.24·14-s + 0.145·15-s − 1.24·16-s + 0.119·17-s − 1.57·18-s + 1.70·19-s + 0.0542·20-s − 2.70·21-s + 0.217·22-s + 1.12·23-s + 0.823·24-s − 0.990·25-s − 0.439·26-s + 0.396·27-s − 1.01·28-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 \( 1 + 3.53T + 8T^{2} \)
3 \( 1 - 7.81T + 27T^{2} \)
5 \( 1 - 1.07T + 125T^{2} \)
7 \( 1 + 33.3T + 343T^{2} \)
11 \( 1 + 6.35T + 1.33e3T^{2} \)
13 \( 1 - 16.4T + 2.19e3T^{2} \)
17 \( 1 - 8.35T + 4.91e3T^{2} \)
19 \( 1 - 141.T + 6.85e3T^{2} \)
23 \( 1 - 123.T + 1.21e4T^{2} \)
29 \( 1 + 54.4T + 2.43e4T^{2} \)
31 \( 1 + 253.T + 2.97e4T^{2} \)
37 \( 1 + 246.T + 5.06e4T^{2} \)
41 \( 1 - 492.T + 6.89e4T^{2} \)
47 \( 1 + 335.T + 1.03e5T^{2} \)
53 \( 1 - 386.T + 1.48e5T^{2} \)
59 \( 1 + 480.T + 2.05e5T^{2} \)
61 \( 1 + 264.T + 2.26e5T^{2} \)
67 \( 1 + 308.T + 3.00e5T^{2} \)
71 \( 1 + 67.5T + 3.57e5T^{2} \)
73 \( 1 - 835.T + 3.89e5T^{2} \)
79 \( 1 + 17.9T + 4.93e5T^{2} \)
83 \( 1 - 1.01e3T + 5.71e5T^{2} \)
89 \( 1 - 1.24e3T + 7.04e5T^{2} \)
97 \( 1 + 287.T + 9.12e5T^{2} \)
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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

−8.870237629761421106344179458144, −7.63325279221160514078912397635, −7.49633332182007519976878584115, −6.48987188992651930986135014780, −5.34609916507873640131744604706, −3.80908608175994369834404235599, −3.27788935247624054278356766556, −2.37396189747235372841493030197, −1.21227170392721501581045563924, 0, 1.21227170392721501581045563924, 2.37396189747235372841493030197, 3.27788935247624054278356766556, 3.80908608175994369834404235599, 5.34609916507873640131744604706, 6.48987188992651930986135014780, 7.49633332182007519976878584115, 7.63325279221160514078912397635, 8.870237629761421106344179458144

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