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  − 2.25·2-s + 4.02·3-s − 2.93·4-s + 4.98·5-s − 9.06·6-s + 5.25·7-s + 24.6·8-s − 10.7·9-s − 11.2·10-s + 25.9·11-s − 11.8·12-s + 14.2·13-s − 11.8·14-s + 20.0·15-s − 31.9·16-s + 20.3·17-s + 24.2·18-s − 2.21·19-s − 14.6·20-s + 21.1·21-s − 58.4·22-s + 39.1·23-s + 99.1·24-s − 100.·25-s − 31.9·26-s − 152.·27-s − 15.3·28-s + ⋯
L(s)  = 1  − 0.795·2-s + 0.775·3-s − 0.366·4-s + 0.446·5-s − 0.616·6-s + 0.283·7-s + 1.08·8-s − 0.399·9-s − 0.355·10-s + 0.711·11-s − 0.284·12-s + 0.303·13-s − 0.225·14-s + 0.345·15-s − 0.499·16-s + 0.290·17-s + 0.317·18-s − 0.0267·19-s − 0.163·20-s + 0.219·21-s − 0.566·22-s + 0.354·23-s + 0.843·24-s − 0.800·25-s − 0.241·26-s − 1.08·27-s − 0.103·28-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 \( 1 + 2.25T + 8T^{2} \)
3 \( 1 - 4.02T + 27T^{2} \)
5 \( 1 - 4.98T + 125T^{2} \)
7 \( 1 - 5.25T + 343T^{2} \)
11 \( 1 - 25.9T + 1.33e3T^{2} \)
13 \( 1 - 14.2T + 2.19e3T^{2} \)
17 \( 1 - 20.3T + 4.91e3T^{2} \)
19 \( 1 + 2.21T + 6.85e3T^{2} \)
23 \( 1 - 39.1T + 1.21e4T^{2} \)
29 \( 1 + 277.T + 2.43e4T^{2} \)
31 \( 1 + 47.2T + 2.97e4T^{2} \)
37 \( 1 + 43.9T + 5.06e4T^{2} \)
41 \( 1 - 98.7T + 6.89e4T^{2} \)
47 \( 1 - 488.T + 1.03e5T^{2} \)
53 \( 1 + 230.T + 1.48e5T^{2} \)
59 \( 1 + 400.T + 2.05e5T^{2} \)
61 \( 1 + 13.3T + 2.26e5T^{2} \)
67 \( 1 + 640.T + 3.00e5T^{2} \)
71 \( 1 + 251.T + 3.57e5T^{2} \)
73 \( 1 - 526.T + 3.89e5T^{2} \)
79 \( 1 - 978.T + 4.93e5T^{2} \)
83 \( 1 + 37.2T + 5.71e5T^{2} \)
89 \( 1 + 437.T + 7.04e5T^{2} \)
97 \( 1 + 1.12e3T + 9.12e5T^{2} \)
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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

−8.632292596035223393551144710134, −7.894474099844757804827408942834, −7.31568717091844830668396688658, −6.08960312152215454543734886045, −5.31456260766652001982968752001, −4.17360406701294702057668202018, −3.41837594665037695884902418735, −2.12746331479052446594116746422, −1.33480025013575151007468640415, 0, 1.33480025013575151007468640415, 2.12746331479052446594116746422, 3.41837594665037695884902418735, 4.17360406701294702057668202018, 5.31456260766652001982968752001, 6.08960312152215454543734886045, 7.31568717091844830668396688658, 7.894474099844757804827408942834, 8.632292596035223393551144710134

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