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  + 0.618·2-s + 9.18·3-s − 7.61·4-s − 19.4·5-s + 5.67·6-s − 15.1·7-s − 9.65·8-s + 57.2·9-s − 12.0·10-s − 7.19·11-s − 69.9·12-s + 36.4·13-s − 9.37·14-s − 178.·15-s + 54.9·16-s + 102.·17-s + 35.4·18-s + 51.6·19-s + 147.·20-s − 139.·21-s − 4.44·22-s − 102.·23-s − 88.6·24-s + 252.·25-s + 22.5·26-s + 277.·27-s + 115.·28-s + ⋯
L(s)  = 1  + 0.218·2-s + 1.76·3-s − 0.952·4-s − 1.73·5-s + 0.386·6-s − 0.818·7-s − 0.426·8-s + 2.12·9-s − 0.379·10-s − 0.197·11-s − 1.68·12-s + 0.777·13-s − 0.178·14-s − 3.06·15-s + 0.858·16-s + 1.45·17-s + 0.463·18-s + 0.624·19-s + 1.65·20-s − 1.44·21-s − 0.0431·22-s − 0.929·23-s − 0.753·24-s + 2.01·25-s + 0.170·26-s + 1.98·27-s + 0.779·28-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 \( 1 - 0.618T + 8T^{2} \)
3 \( 1 - 9.18T + 27T^{2} \)
5 \( 1 + 19.4T + 125T^{2} \)
7 \( 1 + 15.1T + 343T^{2} \)
11 \( 1 + 7.19T + 1.33e3T^{2} \)
13 \( 1 - 36.4T + 2.19e3T^{2} \)
17 \( 1 - 102.T + 4.91e3T^{2} \)
19 \( 1 - 51.6T + 6.85e3T^{2} \)
23 \( 1 + 102.T + 1.21e4T^{2} \)
29 \( 1 - 96.3T + 2.43e4T^{2} \)
31 \( 1 + 157.T + 2.97e4T^{2} \)
37 \( 1 - 233.T + 5.06e4T^{2} \)
41 \( 1 + 39.6T + 6.89e4T^{2} \)
47 \( 1 + 455.T + 1.03e5T^{2} \)
53 \( 1 + 602.T + 1.48e5T^{2} \)
59 \( 1 + 124.T + 2.05e5T^{2} \)
61 \( 1 + 838.T + 2.26e5T^{2} \)
67 \( 1 + 166.T + 3.00e5T^{2} \)
71 \( 1 + 215.T + 3.57e5T^{2} \)
73 \( 1 - 108.T + 3.89e5T^{2} \)
79 \( 1 + 494.T + 4.93e5T^{2} \)
83 \( 1 - 712.T + 5.71e5T^{2} \)
89 \( 1 - 431.T + 7.04e5T^{2} \)
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

−8.256211584929901402663961388958, −7.988827267155409554937269293658, −7.40265848871494218417546571091, −6.16425718099333361545638726540, −4.79661558405863002592720937584, −3.99799082576604234455105584145, −3.29793538254337492832558454419, −3.15646885095891396451289847617, −1.25288227611160936432662562871, 0, 1.25288227611160936432662562871, 3.15646885095891396451289847617, 3.29793538254337492832558454419, 3.99799082576604234455105584145, 4.79661558405863002592720937584, 6.16425718099333361545638726540, 7.40265848871494218417546571091, 7.988827267155409554937269293658, 8.256211584929901402663961388958

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