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  + 3.61·2-s + 2.47·3-s + 5.08·4-s + 17.9·5-s + 8.96·6-s − 10.9·7-s − 10.5·8-s − 20.8·9-s + 65.0·10-s − 60.2·11-s + 12.5·12-s + 83.3·13-s − 39.5·14-s + 44.5·15-s − 78.8·16-s − 55.2·17-s − 75.4·18-s + 24.4·19-s + 91.5·20-s − 27.0·21-s − 217.·22-s − 128.·23-s − 26.1·24-s + 198.·25-s + 301.·26-s − 118.·27-s − 55.5·28-s + ⋯
L(s)  = 1  + 1.27·2-s + 0.476·3-s + 0.635·4-s + 1.60·5-s + 0.609·6-s − 0.589·7-s − 0.465·8-s − 0.772·9-s + 2.05·10-s − 1.65·11-s + 0.303·12-s + 1.77·13-s − 0.754·14-s + 0.767·15-s − 1.23·16-s − 0.788·17-s − 0.988·18-s + 0.294·19-s + 1.02·20-s − 0.281·21-s − 2.11·22-s − 1.16·23-s − 0.222·24-s + 1.59·25-s + 2.27·26-s − 0.845·27-s − 0.375·28-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 \( 1 - 3.61T + 8T^{2} \)
3 \( 1 - 2.47T + 27T^{2} \)
5 \( 1 - 17.9T + 125T^{2} \)
7 \( 1 + 10.9T + 343T^{2} \)
11 \( 1 + 60.2T + 1.33e3T^{2} \)
13 \( 1 - 83.3T + 2.19e3T^{2} \)
17 \( 1 + 55.2T + 4.91e3T^{2} \)
19 \( 1 - 24.4T + 6.85e3T^{2} \)
23 \( 1 + 128.T + 1.21e4T^{2} \)
29 \( 1 + 119.T + 2.43e4T^{2} \)
31 \( 1 + 124.T + 2.97e4T^{2} \)
37 \( 1 - 279.T + 5.06e4T^{2} \)
41 \( 1 - 88.4T + 6.89e4T^{2} \)
47 \( 1 - 102.T + 1.03e5T^{2} \)
53 \( 1 - 184.T + 1.48e5T^{2} \)
59 \( 1 + 539.T + 2.05e5T^{2} \)
61 \( 1 + 709.T + 2.26e5T^{2} \)
67 \( 1 - 74.0T + 3.00e5T^{2} \)
71 \( 1 + 555.T + 3.57e5T^{2} \)
73 \( 1 + 659.T + 3.89e5T^{2} \)
79 \( 1 + 531.T + 4.93e5T^{2} \)
83 \( 1 + 731.T + 5.71e5T^{2} \)
89 \( 1 + 927.T + 7.04e5T^{2} \)
97 \( 1 - 1.29e3T + 9.12e5T^{2} \)
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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

−8.703052003651796915016486563660, −7.65387708640980504031652720215, −6.30061173288517226344427229238, −5.89912752007484110737903385956, −5.51070808406937066273480761090, −4.37015735644705035420564518440, −3.28525318459946483903365882298, −2.69052010096873918916712950262, −1.85904886221934671156244227605, 0, 1.85904886221934671156244227605, 2.69052010096873918916712950262, 3.28525318459946483903365882298, 4.37015735644705035420564518440, 5.51070808406937066273480761090, 5.89912752007484110737903385956, 6.30061173288517226344427229238, 7.65387708640980504031652720215, 8.703052003651796915016486563660

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