Properties

Label 2-43e2-1.1-c3-0-312
Degree $2$
Conductor $1849$
Sign $-1$
Analytic cond. $109.094$
Root an. cond. $10.4448$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.28·2-s + 6.58·3-s − 2.76·4-s − 7.90·5-s − 15.0·6-s + 15.8·7-s + 24.6·8-s + 16.3·9-s + 18.0·10-s − 26.9·11-s − 18.1·12-s + 26.9·13-s − 36.2·14-s − 52.0·15-s − 34.2·16-s − 16.0·17-s − 37.3·18-s + 34.7·19-s + 21.8·20-s + 104.·21-s + 61.6·22-s − 134.·23-s + 162.·24-s − 62.5·25-s − 61.7·26-s − 70.3·27-s − 43.7·28-s + ⋯
L(s)  = 1  − 0.809·2-s + 1.26·3-s − 0.345·4-s − 0.706·5-s − 1.02·6-s + 0.854·7-s + 1.08·8-s + 0.604·9-s + 0.571·10-s − 0.737·11-s − 0.437·12-s + 0.576·13-s − 0.691·14-s − 0.895·15-s − 0.535·16-s − 0.228·17-s − 0.488·18-s + 0.419·19-s + 0.244·20-s + 1.08·21-s + 0.597·22-s − 1.22·23-s + 1.37·24-s − 0.500·25-s − 0.466·26-s − 0.501·27-s − 0.295·28-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: $-1$
Analytic conductor: \(109.094\)
Root analytic conductor: \(10.4448\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
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 \( 1 + 2.28T + 8T^{2} \)
3 \( 1 - 6.58T + 27T^{2} \)
5 \( 1 + 7.90T + 125T^{2} \)
7 \( 1 - 15.8T + 343T^{2} \)
11 \( 1 + 26.9T + 1.33e3T^{2} \)
13 \( 1 - 26.9T + 2.19e3T^{2} \)
17 \( 1 + 16.0T + 4.91e3T^{2} \)
19 \( 1 - 34.7T + 6.85e3T^{2} \)
23 \( 1 + 134.T + 1.21e4T^{2} \)
29 \( 1 - 218.T + 2.43e4T^{2} \)
31 \( 1 - 144.T + 2.97e4T^{2} \)
37 \( 1 - 413.T + 5.06e4T^{2} \)
41 \( 1 + 459.T + 6.89e4T^{2} \)
47 \( 1 + 124.T + 1.03e5T^{2} \)
53 \( 1 - 140.T + 1.48e5T^{2} \)
59 \( 1 + 336.T + 2.05e5T^{2} \)
61 \( 1 + 498.T + 2.26e5T^{2} \)
67 \( 1 + 376.T + 3.00e5T^{2} \)
71 \( 1 + 752.T + 3.57e5T^{2} \)
73 \( 1 - 529.T + 3.89e5T^{2} \)
79 \( 1 - 1.15e3T + 4.93e5T^{2} \)
83 \( 1 + 578.T + 5.71e5T^{2} \)
89 \( 1 - 1.18e3T + 7.04e5T^{2} \)
97 \( 1 - 991.T + 9.12e5T^{2} \)
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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

−8.383320995754240150243830670558, −7.85921212860160041681058530814, −7.70339685809716938056567034739, −6.26698089310286697925407823985, −4.93658421950533059721916257177, −4.27052623549983359969786924216, −3.36681465779116687273263161757, −2.29126668680598379235681788948, −1.26814718889861167715385597466, 0, 1.26814718889861167715385597466, 2.29126668680598379235681788948, 3.36681465779116687273263161757, 4.27052623549983359969786924216, 4.93658421950533059721916257177, 6.26698089310286697925407823985, 7.70339685809716938056567034739, 7.85921212860160041681058530814, 8.383320995754240150243830670558

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