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  − 4.85·2-s − 5.73·3-s + 15.6·4-s + 14.4·5-s + 27.8·6-s − 13.1·7-s − 36.9·8-s + 5.83·9-s − 70.2·10-s − 56.6·11-s − 89.4·12-s − 68.5·13-s + 64.0·14-s − 82.8·15-s + 54.6·16-s + 23.3·17-s − 28.3·18-s − 119.·19-s + 225.·20-s + 75.5·21-s + 275.·22-s − 49.8·23-s + 211.·24-s + 84.0·25-s + 333.·26-s + 121.·27-s − 205.·28-s + ⋯
L(s)  = 1  − 1.71·2-s − 1.10·3-s + 1.95·4-s + 1.29·5-s + 1.89·6-s − 0.711·7-s − 1.63·8-s + 0.216·9-s − 2.22·10-s − 1.55·11-s − 2.15·12-s − 1.46·13-s + 1.22·14-s − 1.42·15-s + 0.854·16-s + 0.333·17-s − 0.371·18-s − 1.44·19-s + 2.52·20-s + 0.785·21-s + 2.66·22-s − 0.451·23-s + 1.80·24-s + 0.672·25-s + 2.51·26-s + 0.864·27-s − 1.38·28-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 \( 1 + 4.85T + 8T^{2} \)
3 \( 1 + 5.73T + 27T^{2} \)
5 \( 1 - 14.4T + 125T^{2} \)
7 \( 1 + 13.1T + 343T^{2} \)
11 \( 1 + 56.6T + 1.33e3T^{2} \)
13 \( 1 + 68.5T + 2.19e3T^{2} \)
17 \( 1 - 23.3T + 4.91e3T^{2} \)
19 \( 1 + 119.T + 6.85e3T^{2} \)
23 \( 1 + 49.8T + 1.21e4T^{2} \)
29 \( 1 - 290.T + 2.43e4T^{2} \)
31 \( 1 - 58.9T + 2.97e4T^{2} \)
37 \( 1 - 147.T + 5.06e4T^{2} \)
41 \( 1 - 216.T + 6.89e4T^{2} \)
47 \( 1 + 233.T + 1.03e5T^{2} \)
53 \( 1 - 712.T + 1.48e5T^{2} \)
59 \( 1 - 70.0T + 2.05e5T^{2} \)
61 \( 1 - 328.T + 2.26e5T^{2} \)
67 \( 1 - 307.T + 3.00e5T^{2} \)
71 \( 1 - 1.13e3T + 3.57e5T^{2} \)
73 \( 1 - 203.T + 3.89e5T^{2} \)
79 \( 1 - 255.T + 4.93e5T^{2} \)
83 \( 1 - 665.T + 5.71e5T^{2} \)
89 \( 1 + 834.T + 7.04e5T^{2} \)
97 \( 1 - 817.T + 9.12e5T^{2} \)
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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

−8.526202283828307679037679336255, −7.88764782848712559633596191825, −6.80458645042228078414721689677, −6.36760930326932259365738129419, −5.55306892885260627510543812949, −4.77512763780250061682941365325, −2.58132158635269619740504580913, −2.31926382920898895344640621249, −0.77813855566844007094210072974, 0, 0.77813855566844007094210072974, 2.31926382920898895344640621249, 2.58132158635269619740504580913, 4.77512763780250061682941365325, 5.55306892885260627510543812949, 6.36760930326932259365738129419, 6.80458645042228078414721689677, 7.88764782848712559633596191825, 8.526202283828307679037679336255

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