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  + 4.55·2-s + 4.29·3-s + 12.7·4-s + 2.70·5-s + 19.5·6-s − 9.48·7-s + 21.5·8-s − 8.58·9-s + 12.2·10-s − 16.2·11-s + 54.6·12-s − 58.6·13-s − 43.1·14-s + 11.5·15-s − 3.68·16-s + 67.1·17-s − 39.1·18-s − 41.5·19-s + 34.3·20-s − 40.6·21-s − 74.0·22-s − 85.8·23-s + 92.5·24-s − 117.·25-s − 266.·26-s − 152.·27-s − 120.·28-s + ⋯
L(s)  = 1  + 1.60·2-s + 0.825·3-s + 1.59·4-s + 0.241·5-s + 1.32·6-s − 0.511·7-s + 0.953·8-s − 0.318·9-s + 0.388·10-s − 0.445·11-s + 1.31·12-s − 1.25·13-s − 0.824·14-s + 0.199·15-s − 0.0575·16-s + 0.958·17-s − 0.512·18-s − 0.501·19-s + 0.384·20-s − 0.422·21-s − 0.717·22-s − 0.778·23-s + 0.787·24-s − 0.941·25-s − 2.01·26-s − 1.08·27-s − 0.814·28-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 \( 1 - 4.55T + 8T^{2} \)
3 \( 1 - 4.29T + 27T^{2} \)
5 \( 1 - 2.70T + 125T^{2} \)
7 \( 1 + 9.48T + 343T^{2} \)
11 \( 1 + 16.2T + 1.33e3T^{2} \)
13 \( 1 + 58.6T + 2.19e3T^{2} \)
17 \( 1 - 67.1T + 4.91e3T^{2} \)
19 \( 1 + 41.5T + 6.85e3T^{2} \)
23 \( 1 + 85.8T + 1.21e4T^{2} \)
29 \( 1 + 162.T + 2.43e4T^{2} \)
31 \( 1 - 270.T + 2.97e4T^{2} \)
37 \( 1 - 226.T + 5.06e4T^{2} \)
41 \( 1 + 360.T + 6.89e4T^{2} \)
47 \( 1 - 437.T + 1.03e5T^{2} \)
53 \( 1 + 377.T + 1.48e5T^{2} \)
59 \( 1 + 279.T + 2.05e5T^{2} \)
61 \( 1 + 30.6T + 2.26e5T^{2} \)
67 \( 1 - 904.T + 3.00e5T^{2} \)
71 \( 1 - 121.T + 3.57e5T^{2} \)
73 \( 1 - 372.T + 3.89e5T^{2} \)
79 \( 1 - 730.T + 4.93e5T^{2} \)
83 \( 1 + 73.2T + 5.71e5T^{2} \)
89 \( 1 + 1.15e3T + 7.04e5T^{2} \)
97 \( 1 + 446.T + 9.12e5T^{2} \)
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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

−8.231733132378221086620851383148, −7.66636326117869998887348919097, −6.65050035577996033423118010003, −5.86256752583284328876432397644, −5.22741198777056320521672949721, −4.25580420976649388363378624709, −3.42544298785772667150026071319, −2.69088608700245749848716873544, −2.05050491443045321446278788185, 0, 2.05050491443045321446278788185, 2.69088608700245749848716873544, 3.42544298785772667150026071319, 4.25580420976649388363378624709, 5.22741198777056320521672949721, 5.86256752583284328876432397644, 6.65050035577996033423118010003, 7.66636326117869998887348919097, 8.231733132378221086620851383148

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