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  + 1.06·2-s + 0.463·3-s − 6.86·4-s + 5.41·5-s + 0.493·6-s + 4.55·7-s − 15.8·8-s − 26.7·9-s + 5.76·10-s + 10.2·11-s − 3.18·12-s + 4.48·13-s + 4.85·14-s + 2.50·15-s + 38.0·16-s − 32.2·17-s − 28.5·18-s + 54.9·19-s − 37.1·20-s + 2.11·21-s + 10.9·22-s + 136.·23-s − 7.34·24-s − 95.7·25-s + 4.78·26-s − 24.9·27-s − 31.2·28-s + ⋯
L(s)  = 1  + 0.376·2-s + 0.0892·3-s − 0.858·4-s + 0.484·5-s + 0.0335·6-s + 0.246·7-s − 0.699·8-s − 0.992·9-s + 0.182·10-s + 0.281·11-s − 0.0765·12-s + 0.0957·13-s + 0.0926·14-s + 0.0431·15-s + 0.594·16-s − 0.459·17-s − 0.373·18-s + 0.663·19-s − 0.415·20-s + 0.0219·21-s + 0.106·22-s + 1.23·23-s − 0.0624·24-s − 0.765·25-s + 0.0360·26-s − 0.177·27-s − 0.211·28-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 \( 1 - 1.06T + 8T^{2} \)
3 \( 1 - 0.463T + 27T^{2} \)
5 \( 1 - 5.41T + 125T^{2} \)
7 \( 1 - 4.55T + 343T^{2} \)
11 \( 1 - 10.2T + 1.33e3T^{2} \)
13 \( 1 - 4.48T + 2.19e3T^{2} \)
17 \( 1 + 32.2T + 4.91e3T^{2} \)
19 \( 1 - 54.9T + 6.85e3T^{2} \)
23 \( 1 - 136.T + 1.21e4T^{2} \)
29 \( 1 - 87.3T + 2.43e4T^{2} \)
31 \( 1 - 27.1T + 2.97e4T^{2} \)
37 \( 1 + 287.T + 5.06e4T^{2} \)
41 \( 1 - 285.T + 6.89e4T^{2} \)
47 \( 1 + 62.1T + 1.03e5T^{2} \)
53 \( 1 - 310.T + 1.48e5T^{2} \)
59 \( 1 - 561.T + 2.05e5T^{2} \)
61 \( 1 + 381.T + 2.26e5T^{2} \)
67 \( 1 + 167.T + 3.00e5T^{2} \)
71 \( 1 + 328.T + 3.57e5T^{2} \)
73 \( 1 + 1.03e3T + 3.89e5T^{2} \)
79 \( 1 + 496.T + 4.93e5T^{2} \)
83 \( 1 - 374.T + 5.71e5T^{2} \)
89 \( 1 - 270.T + 7.04e5T^{2} \)
97 \( 1 + 1.11e3T + 9.12e5T^{2} \)
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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

−8.702601303718258014228479265306, −7.85344166303441386076594212233, −6.77177597049604941457571539553, −5.83246084588617631852616505061, −5.28633063590817437606438724267, −4.44186848686654251510291441760, −3.44039389913820384900366604795, −2.61303449852485078509689438226, −1.23657124882801991139565804877, 0, 1.23657124882801991139565804877, 2.61303449852485078509689438226, 3.44039389913820384900366604795, 4.44186848686654251510291441760, 5.28633063590817437606438724267, 5.83246084588617631852616505061, 6.77177597049604941457571539553, 7.85344166303441386076594212233, 8.702601303718258014228479265306

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