Properties

Label 2-43e2-1.1-c3-0-337
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  − 4.36·2-s + 4.55·3-s + 11.0·4-s + 9.21·5-s − 19.9·6-s + 12.2·7-s − 13.3·8-s − 6.22·9-s − 40.2·10-s + 16.8·11-s + 50.4·12-s − 74.1·13-s − 53.5·14-s + 42.0·15-s − 30.1·16-s + 0.711·17-s + 27.1·18-s − 31.2·19-s + 101.·20-s + 55.9·21-s − 73.4·22-s + 83.8·23-s − 60.9·24-s − 40.0·25-s + 323.·26-s − 151.·27-s + 135.·28-s + ⋯
L(s)  = 1  − 1.54·2-s + 0.877·3-s + 1.38·4-s + 0.824·5-s − 1.35·6-s + 0.662·7-s − 0.590·8-s − 0.230·9-s − 1.27·10-s + 0.460·11-s + 1.21·12-s − 1.58·13-s − 1.02·14-s + 0.722·15-s − 0.470·16-s + 0.0101·17-s + 0.355·18-s − 0.377·19-s + 1.13·20-s + 0.581·21-s − 0.711·22-s + 0.760·23-s − 0.518·24-s − 0.320·25-s + 2.44·26-s − 1.07·27-s + 0.915·28-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 \( 1 + 4.36T + 8T^{2} \)
3 \( 1 - 4.55T + 27T^{2} \)
5 \( 1 - 9.21T + 125T^{2} \)
7 \( 1 - 12.2T + 343T^{2} \)
11 \( 1 - 16.8T + 1.33e3T^{2} \)
13 \( 1 + 74.1T + 2.19e3T^{2} \)
17 \( 1 - 0.711T + 4.91e3T^{2} \)
19 \( 1 + 31.2T + 6.85e3T^{2} \)
23 \( 1 - 83.8T + 1.21e4T^{2} \)
29 \( 1 - 228.T + 2.43e4T^{2} \)
31 \( 1 + 243.T + 2.97e4T^{2} \)
37 \( 1 + 82.4T + 5.06e4T^{2} \)
41 \( 1 - 404.T + 6.89e4T^{2} \)
47 \( 1 - 350.T + 1.03e5T^{2} \)
53 \( 1 - 106.T + 1.48e5T^{2} \)
59 \( 1 + 684.T + 2.05e5T^{2} \)
61 \( 1 + 505.T + 2.26e5T^{2} \)
67 \( 1 - 286.T + 3.00e5T^{2} \)
71 \( 1 - 131.T + 3.57e5T^{2} \)
73 \( 1 + 728.T + 3.89e5T^{2} \)
79 \( 1 - 818.T + 4.93e5T^{2} \)
83 \( 1 + 866.T + 5.71e5T^{2} \)
89 \( 1 + 875.T + 7.04e5T^{2} \)
97 \( 1 + 931.T + 9.12e5T^{2} \)
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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

−8.745793981233602089441866591358, −7.81037406943078170931597765423, −7.39851053971565422823488033514, −6.42096170829835104163977945612, −5.34074059380119945834410482397, −4.33797119726269403134177995651, −2.80881842253526311237934320714, −2.18771672004516125943104082745, −1.34955998870961236611234688535, 0, 1.34955998870961236611234688535, 2.18771672004516125943104082745, 2.80881842253526311237934320714, 4.33797119726269403134177995651, 5.34074059380119945834410482397, 6.42096170829835104163977945612, 7.39851053971565422823488033514, 7.81037406943078170931597765423, 8.745793981233602089441866591358

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