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  − 1.89·2-s + 6.15·3-s − 4.39·4-s + 1.87·5-s − 11.6·6-s − 2.77·7-s + 23.5·8-s + 10.8·9-s − 3.57·10-s + 5.01·11-s − 27.0·12-s − 31.2·13-s + 5.27·14-s + 11.5·15-s − 9.56·16-s + 42.4·17-s − 20.5·18-s − 2.31·19-s − 8.25·20-s − 17.0·21-s − 9.52·22-s − 10.8·23-s + 144.·24-s − 121.·25-s + 59.4·26-s − 99.3·27-s + 12.2·28-s + ⋯
L(s)  = 1  − 0.671·2-s + 1.18·3-s − 0.549·4-s + 0.168·5-s − 0.795·6-s − 0.149·7-s + 1.04·8-s + 0.401·9-s − 0.112·10-s + 0.137·11-s − 0.650·12-s − 0.667·13-s + 0.100·14-s + 0.199·15-s − 0.149·16-s + 0.605·17-s − 0.269·18-s − 0.0279·19-s − 0.0923·20-s − 0.177·21-s − 0.0922·22-s − 0.0983·23-s + 1.23·24-s − 0.971·25-s + 0.448·26-s − 0.708·27-s + 0.0823·28-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 \( 1 + 1.89T + 8T^{2} \)
3 \( 1 - 6.15T + 27T^{2} \)
5 \( 1 - 1.87T + 125T^{2} \)
7 \( 1 + 2.77T + 343T^{2} \)
11 \( 1 - 5.01T + 1.33e3T^{2} \)
13 \( 1 + 31.2T + 2.19e3T^{2} \)
17 \( 1 - 42.4T + 4.91e3T^{2} \)
19 \( 1 + 2.31T + 6.85e3T^{2} \)
23 \( 1 + 10.8T + 1.21e4T^{2} \)
29 \( 1 - 82.1T + 2.43e4T^{2} \)
31 \( 1 - 296.T + 2.97e4T^{2} \)
37 \( 1 + 186.T + 5.06e4T^{2} \)
41 \( 1 - 343.T + 6.89e4T^{2} \)
47 \( 1 + 200.T + 1.03e5T^{2} \)
53 \( 1 + 500.T + 1.48e5T^{2} \)
59 \( 1 - 613.T + 2.05e5T^{2} \)
61 \( 1 + 661.T + 2.26e5T^{2} \)
67 \( 1 - 198.T + 3.00e5T^{2} \)
71 \( 1 - 896.T + 3.57e5T^{2} \)
73 \( 1 + 619.T + 3.89e5T^{2} \)
79 \( 1 - 58.4T + 4.93e5T^{2} \)
83 \( 1 - 188.T + 5.71e5T^{2} \)
89 \( 1 + 1.32e3T + 7.04e5T^{2} \)
97 \( 1 - 1.02e3T + 9.12e5T^{2} \)
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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

−8.396783978386241156138008580982, −8.049620548251523040973607331734, −7.31543306509800747193428762389, −6.21492441143341400101994807502, −5.10395368336692350281753278364, −4.22745139743161632100183506217, −3.30080491788353717699647234486, −2.36752960773292475755280876302, −1.29205792415426080888398329199, 0, 1.29205792415426080888398329199, 2.36752960773292475755280876302, 3.30080491788353717699647234486, 4.22745139743161632100183506217, 5.10395368336692350281753278364, 6.21492441143341400101994807502, 7.31543306509800747193428762389, 8.049620548251523040973607331734, 8.396783978386241156138008580982

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