Properties

Label 2-43e2-1.1-c3-0-357
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  + 1.56·2-s + 5.42·3-s − 5.54·4-s − 8.12·5-s + 8.49·6-s + 30.7·7-s − 21.2·8-s + 2.42·9-s − 12.7·10-s + 9.06·11-s − 30.0·12-s − 16.1·13-s + 48.1·14-s − 44.0·15-s + 11.1·16-s − 13.4·17-s + 3.79·18-s + 76.7·19-s + 45.0·20-s + 166.·21-s + 14.1·22-s − 38.4·23-s − 115.·24-s − 58.9·25-s − 25.2·26-s − 133.·27-s − 170.·28-s + ⋯
L(s)  = 1  + 0.553·2-s + 1.04·3-s − 0.693·4-s − 0.726·5-s + 0.578·6-s + 1.65·7-s − 0.937·8-s + 0.0896·9-s − 0.402·10-s + 0.248·11-s − 0.723·12-s − 0.343·13-s + 0.918·14-s − 0.758·15-s + 0.174·16-s − 0.191·17-s + 0.0496·18-s + 0.926·19-s + 0.503·20-s + 1.73·21-s + 0.137·22-s − 0.348·23-s − 0.978·24-s − 0.471·25-s − 0.190·26-s − 0.950·27-s − 1.15·28-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 \( 1 - 1.56T + 8T^{2} \)
3 \( 1 - 5.42T + 27T^{2} \)
5 \( 1 + 8.12T + 125T^{2} \)
7 \( 1 - 30.7T + 343T^{2} \)
11 \( 1 - 9.06T + 1.33e3T^{2} \)
13 \( 1 + 16.1T + 2.19e3T^{2} \)
17 \( 1 + 13.4T + 4.91e3T^{2} \)
19 \( 1 - 76.7T + 6.85e3T^{2} \)
23 \( 1 + 38.4T + 1.21e4T^{2} \)
29 \( 1 + 251.T + 2.43e4T^{2} \)
31 \( 1 + 120.T + 2.97e4T^{2} \)
37 \( 1 - 391.T + 5.06e4T^{2} \)
41 \( 1 + 338.T + 6.89e4T^{2} \)
47 \( 1 - 120.T + 1.03e5T^{2} \)
53 \( 1 + 480.T + 1.48e5T^{2} \)
59 \( 1 - 357.T + 2.05e5T^{2} \)
61 \( 1 - 592.T + 2.26e5T^{2} \)
67 \( 1 + 672.T + 3.00e5T^{2} \)
71 \( 1 + 417.T + 3.57e5T^{2} \)
73 \( 1 - 922.T + 3.89e5T^{2} \)
79 \( 1 + 107.T + 4.93e5T^{2} \)
83 \( 1 + 1.00e3T + 5.71e5T^{2} \)
89 \( 1 - 188.T + 7.04e5T^{2} \)
97 \( 1 - 481.T + 9.12e5T^{2} \)
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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

−8.255356627873117393990768723224, −7.996598355503361225325330531150, −7.25437328776790128488458894834, −5.73080931044441212014274351487, −5.09572901587643237039521520149, −4.15355707437447970510128639277, −3.69127918419213235544374961480, −2.58662104705432730876653589949, −1.49131091326860107514086242271, 0, 1.49131091326860107514086242271, 2.58662104705432730876653589949, 3.69127918419213235544374961480, 4.15355707437447970510128639277, 5.09572901587643237039521520149, 5.73080931044441212014274351487, 7.25437328776790128488458894834, 7.996598355503361225325330531150, 8.255356627873117393990768723224

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