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  − 3.52·2-s − 3.06·3-s + 4.44·4-s − 14.8·5-s + 10.8·6-s − 18.1·7-s + 12.5·8-s − 17.6·9-s + 52.4·10-s − 54.4·11-s − 13.6·12-s − 71.7·13-s + 64.0·14-s + 45.5·15-s − 79.7·16-s − 70.3·17-s + 62.1·18-s − 71.5·19-s − 66.1·20-s + 55.6·21-s + 192.·22-s + 107.·23-s − 38.3·24-s + 95.6·25-s + 253.·26-s + 136.·27-s − 80.8·28-s + ⋯
L(s)  = 1  − 1.24·2-s − 0.589·3-s + 0.556·4-s − 1.32·5-s + 0.735·6-s − 0.980·7-s + 0.553·8-s − 0.652·9-s + 1.65·10-s − 1.49·11-s − 0.327·12-s − 1.53·13-s + 1.22·14-s + 0.783·15-s − 1.24·16-s − 1.00·17-s + 0.813·18-s − 0.863·19-s − 0.739·20-s + 0.578·21-s + 1.86·22-s + 0.977·23-s − 0.326·24-s + 0.765·25-s + 1.90·26-s + 0.974·27-s − 0.545·28-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 \( 1 + 3.52T + 8T^{2} \)
3 \( 1 + 3.06T + 27T^{2} \)
5 \( 1 + 14.8T + 125T^{2} \)
7 \( 1 + 18.1T + 343T^{2} \)
11 \( 1 + 54.4T + 1.33e3T^{2} \)
13 \( 1 + 71.7T + 2.19e3T^{2} \)
17 \( 1 + 70.3T + 4.91e3T^{2} \)
19 \( 1 + 71.5T + 6.85e3T^{2} \)
23 \( 1 - 107.T + 1.21e4T^{2} \)
29 \( 1 - 165.T + 2.43e4T^{2} \)
31 \( 1 + 117.T + 2.97e4T^{2} \)
37 \( 1 + 37.0T + 5.06e4T^{2} \)
41 \( 1 - 24.0T + 6.89e4T^{2} \)
47 \( 1 + 454.T + 1.03e5T^{2} \)
53 \( 1 + 662.T + 1.48e5T^{2} \)
59 \( 1 - 325.T + 2.05e5T^{2} \)
61 \( 1 + 545.T + 2.26e5T^{2} \)
67 \( 1 - 976.T + 3.00e5T^{2} \)
71 \( 1 + 494.T + 3.57e5T^{2} \)
73 \( 1 - 462.T + 3.89e5T^{2} \)
79 \( 1 - 51.1T + 4.93e5T^{2} \)
83 \( 1 - 391.T + 5.71e5T^{2} \)
89 \( 1 - 1.09e3T + 7.04e5T^{2} \)
97 \( 1 + 685.T + 9.12e5T^{2} \)
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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

−8.380397301355703156422898105223, −7.913522160987714541447514158387, −7.08633529017440689678866156485, −6.48630731193827428618822754030, −5.06364031291904481526711965028, −4.59608677904755733133539153125, −3.18302970866473615654784245873, −2.33926375503586171528390461877, −0.41663249541034762289167925040, 0, 0.41663249541034762289167925040, 2.33926375503586171528390461877, 3.18302970866473615654784245873, 4.59608677904755733133539153125, 5.06364031291904481526711965028, 6.48630731193827428618822754030, 7.08633529017440689678866156485, 7.913522160987714541447514158387, 8.380397301355703156422898105223

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