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  + 4.08·2-s + 6.02·3-s + 8.64·4-s − 2.18·5-s + 24.5·6-s − 8.90·7-s + 2.64·8-s + 9.26·9-s − 8.90·10-s + 2.96·11-s + 52.0·12-s + 15.9·13-s − 36.3·14-s − 13.1·15-s − 58.3·16-s − 54.4·17-s + 37.8·18-s − 136.·19-s − 18.8·20-s − 53.6·21-s + 12.0·22-s + 153.·23-s + 15.9·24-s − 120.·25-s + 64.9·26-s − 106.·27-s − 77.0·28-s + ⋯
L(s)  = 1  + 1.44·2-s + 1.15·3-s + 1.08·4-s − 0.195·5-s + 1.67·6-s − 0.480·7-s + 0.117·8-s + 0.343·9-s − 0.281·10-s + 0.0811·11-s + 1.25·12-s + 0.339·13-s − 0.693·14-s − 0.226·15-s − 0.912·16-s − 0.776·17-s + 0.495·18-s − 1.64·19-s − 0.211·20-s − 0.557·21-s + 0.117·22-s + 1.38·23-s + 0.135·24-s − 0.961·25-s + 0.490·26-s − 0.761·27-s − 0.519·28-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 \( 1 - 4.08T + 8T^{2} \)
3 \( 1 - 6.02T + 27T^{2} \)
5 \( 1 + 2.18T + 125T^{2} \)
7 \( 1 + 8.90T + 343T^{2} \)
11 \( 1 - 2.96T + 1.33e3T^{2} \)
13 \( 1 - 15.9T + 2.19e3T^{2} \)
17 \( 1 + 54.4T + 4.91e3T^{2} \)
19 \( 1 + 136.T + 6.85e3T^{2} \)
23 \( 1 - 153.T + 1.21e4T^{2} \)
29 \( 1 + 69.4T + 2.43e4T^{2} \)
31 \( 1 + 13.1T + 2.97e4T^{2} \)
37 \( 1 - 306.T + 5.06e4T^{2} \)
41 \( 1 + 27.1T + 6.89e4T^{2} \)
47 \( 1 + 621.T + 1.03e5T^{2} \)
53 \( 1 + 97.9T + 1.48e5T^{2} \)
59 \( 1 + 417.T + 2.05e5T^{2} \)
61 \( 1 - 608.T + 2.26e5T^{2} \)
67 \( 1 - 772.T + 3.00e5T^{2} \)
71 \( 1 + 858.T + 3.57e5T^{2} \)
73 \( 1 + 14.7T + 3.89e5T^{2} \)
79 \( 1 - 872.T + 4.93e5T^{2} \)
83 \( 1 + 1.05e3T + 5.71e5T^{2} \)
89 \( 1 - 444.T + 7.04e5T^{2} \)
97 \( 1 - 150.T + 9.12e5T^{2} \)
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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

−8.560975303613695935665998105168, −7.66347099449586885156043229104, −6.60580960592821299638145506176, −6.14569443214677607259695572476, −4.98958983188124621798165916354, −4.16118737418352093522116964306, −3.51782998668207761130934606913, −2.75260688608936636862692473897, −1.94910646847027081673383525749, 0, 1.94910646847027081673383525749, 2.75260688608936636862692473897, 3.51782998668207761130934606913, 4.16118737418352093522116964306, 4.98958983188124621798165916354, 6.14569443214677607259695572476, 6.60580960592821299638145506176, 7.66347099449586885156043229104, 8.560975303613695935665998105168

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