↑

Properties

Label 2-379050-1.1-c1-0-161
Degree $2$
Conductor $379050$
Sign $-1$
Analytic cond. $3026.72$
Root an. cond. $55.0157$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 12-s + 4·13-s + 14-s + 16-s + 8·17-s − 18-s + 21-s − 2·23-s + 24-s − 4·26-s − 27-s − 28-s − 6·29-s + 6·31-s − 32-s − 8·34-s + 36-s − 2·37-s − 4·39-s − 6·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s − 0.235·18-s + 0.218·21-s − 0.417·23-s + 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 1.07·31-s − 0.176·32-s − 1.37·34-s + 1/6·36-s − 0.328·37-s − 0.640·39-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(379050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(3026.72\)
Root analytic conductor: \(55.0157\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
19 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−12.71194749677774, −11.96406973917541, −11.72421466569102, −11.54542903186870, −10.71807092649516, −10.35597103245242, −10.13660481857202, −9.568204080027201, −9.216465814746818, −8.423456750772887, −8.300551467799226, −7.710657142905767, −7.201572362538010, −6.741322284306811, −6.238191941777251, −5.832167480320666, −5.348279834225926, −4.979420964741809, −3.917707355827159, −3.786263774378182, −3.163681137873718, −2.565411839608662, −1.751260515495338, −1.277462206370357, −0.7610979081972580, 0, 0.7610979081972580, 1.277462206370357, 1.751260515495338, 2.565411839608662, 3.163681137873718, 3.786263774378182, 3.917707355827159, 4.979420964741809, 5.348279834225926, 5.832167480320666, 6.238191941777251, 6.741322284306811, 7.201572362538010, 7.710657142905767, 8.300551467799226, 8.423456750772887, 9.216465814746818, 9.568204080027201, 10.13660481857202, 10.35597103245242, 10.71807092649516, 11.54542903186870, 11.72421466569102, 11.96406973917541, 12.71194749677774

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