Properties

Label 2-43e2-1.1-c3-0-214
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  − 5.00·2-s − 7.79·3-s + 17.0·4-s − 17.6·5-s + 39.0·6-s + 25.3·7-s − 45.1·8-s + 33.8·9-s + 88.1·10-s + 2.66·11-s − 132.·12-s + 61.8·13-s − 126.·14-s + 137.·15-s + 89.7·16-s − 11.9·17-s − 169.·18-s + 74.4·19-s − 300.·20-s − 197.·21-s − 13.3·22-s + 55.2·23-s + 352.·24-s + 185.·25-s − 309.·26-s − 53.2·27-s + 431.·28-s + ⋯
L(s)  = 1  − 1.76·2-s − 1.50·3-s + 2.12·4-s − 1.57·5-s + 2.65·6-s + 1.36·7-s − 1.99·8-s + 1.25·9-s + 2.78·10-s + 0.0731·11-s − 3.19·12-s + 1.32·13-s − 2.41·14-s + 2.36·15-s + 1.40·16-s − 0.170·17-s − 2.21·18-s + 0.899·19-s − 3.35·20-s − 2.05·21-s − 0.129·22-s + 0.501·23-s + 2.99·24-s + 1.48·25-s − 2.33·26-s − 0.379·27-s + 2.91·28-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: Trivial
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 \( 1 + 5.00T + 8T^{2} \)
3 \( 1 + 7.79T + 27T^{2} \)
5 \( 1 + 17.6T + 125T^{2} \)
7 \( 1 - 25.3T + 343T^{2} \)
11 \( 1 - 2.66T + 1.33e3T^{2} \)
13 \( 1 - 61.8T + 2.19e3T^{2} \)
17 \( 1 + 11.9T + 4.91e3T^{2} \)
19 \( 1 - 74.4T + 6.85e3T^{2} \)
23 \( 1 - 55.2T + 1.21e4T^{2} \)
29 \( 1 + 26.4T + 2.43e4T^{2} \)
31 \( 1 - 132.T + 2.97e4T^{2} \)
37 \( 1 + 300.T + 5.06e4T^{2} \)
41 \( 1 + 309.T + 6.89e4T^{2} \)
47 \( 1 + 222.T + 1.03e5T^{2} \)
53 \( 1 + 513.T + 1.48e5T^{2} \)
59 \( 1 - 192.T + 2.05e5T^{2} \)
61 \( 1 - 349.T + 2.26e5T^{2} \)
67 \( 1 + 318.T + 3.00e5T^{2} \)
71 \( 1 + 278.T + 3.57e5T^{2} \)
73 \( 1 + 265.T + 3.89e5T^{2} \)
79 \( 1 + 131.T + 4.93e5T^{2} \)
83 \( 1 - 1.48e3T + 5.71e5T^{2} \)
89 \( 1 - 837.T + 7.04e5T^{2} \)
97 \( 1 - 259.T + 9.12e5T^{2} \)
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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

−8.297211630177156812573913743534, −7.958138841440762296657284236214, −7.08304229262354866753416764014, −6.50070090071183769027055603993, −5.33477566736436808826044216992, −4.54136856553582888364395409418, −3.36167543842206158954189174292, −1.55617789631986294369856219744, −0.916957442219643989209880013017, 0, 0.916957442219643989209880013017, 1.55617789631986294369856219744, 3.36167543842206158954189174292, 4.54136856553582888364395409418, 5.33477566736436808826044216992, 6.50070090071183769027055603993, 7.08304229262354866753416764014, 7.958138841440762296657284236214, 8.297211630177156812573913743534

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