Properties

Label 2-8043-1.1-c1-0-173
Degree $2$
Conductor $8043$
Sign $-1$
Analytic cond. $64.2236$
Root an. cond. $8.01396$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.55·2-s + 3-s + 4.52·4-s − 2.76·5-s − 2.55·6-s − 7-s − 6.44·8-s + 9-s + 7.05·10-s − 5.11·11-s + 4.52·12-s + 1.44·13-s + 2.55·14-s − 2.76·15-s + 7.41·16-s + 2.55·17-s − 2.55·18-s + 2.24·19-s − 12.4·20-s − 21-s + 13.0·22-s − 0.475·23-s − 6.44·24-s + 2.62·25-s − 3.69·26-s + 27-s − 4.52·28-s + ⋯
L(s)  = 1  − 1.80·2-s + 0.577·3-s + 2.26·4-s − 1.23·5-s − 1.04·6-s − 0.377·7-s − 2.27·8-s + 0.333·9-s + 2.22·10-s − 1.54·11-s + 1.30·12-s + 0.400·13-s + 0.682·14-s − 0.712·15-s + 1.85·16-s + 0.618·17-s − 0.602·18-s + 0.515·19-s − 2.79·20-s − 0.218·21-s + 2.78·22-s − 0.0991·23-s − 1.31·24-s + 0.524·25-s − 0.723·26-s + 0.192·27-s − 0.854·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8043\)    =    \(3 \cdot 7 \cdot 383\)
Sign: $-1$
Analytic conductor: \(64.2236\)
Root analytic conductor: \(8.01396\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8043,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + T \)
383 \( 1 - T \)
good2 \( 1 + 2.55T + 2T^{2} \)
5 \( 1 + 2.76T + 5T^{2} \)
11 \( 1 + 5.11T + 11T^{2} \)
13 \( 1 - 1.44T + 13T^{2} \)
17 \( 1 - 2.55T + 17T^{2} \)
19 \( 1 - 2.24T + 19T^{2} \)
23 \( 1 + 0.475T + 23T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 - 2.73T + 31T^{2} \)
37 \( 1 - 4.97T + 37T^{2} \)
41 \( 1 - 4.10T + 41T^{2} \)
43 \( 1 + 0.613T + 43T^{2} \)
47 \( 1 - 8.69T + 47T^{2} \)
53 \( 1 + 9.83T + 53T^{2} \)
59 \( 1 + 0.0375T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 + 4.09T + 67T^{2} \)
71 \( 1 + 6.35T + 71T^{2} \)
73 \( 1 - 9.02T + 73T^{2} \)
79 \( 1 - 5.91T + 79T^{2} \)
83 \( 1 - 3.68T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.60393732107773153769228464326, −7.54479412649844682156168922786, −6.43402601689197882014194676442, −5.65402892881226172089467007768, −4.54267495283016364644481298062, −3.49174356223353141065976143761, −2.93476615795673934534173811184, −2.08693115722404674858161744274, −0.913404129704081538507192064988, 0, 0.913404129704081538507192064988, 2.08693115722404674858161744274, 2.93476615795673934534173811184, 3.49174356223353141065976143761, 4.54267495283016364644481298062, 5.65402892881226172089467007768, 6.43402601689197882014194676442, 7.54479412649844682156168922786, 7.60393732107773153769228464326

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