↑

Properties

Label 2-379050-1.1-c1-0-118
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 − 3·11-s − 12-s + 13-s + 14-s + 16-s − 7·17-s − 18-s + 21-s + 3·22-s + 7·23-s + 24-s − 26-s − 27-s − 28-s − 3·29-s − 32-s + 3·33-s + 7·34-s + 36-s + 4·37-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 − 0.904·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.69·17-s − 0.235·18-s + 0.218·21-s + 0.639·22-s + 1.45·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.188·28-s − 0.557·29-s − 0.176·32-s + 0.522·33-s + 1.20·34-s + 1/6·36-s + 0.657·37-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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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

−12.77967423150328, −12.20916129536892, −11.57290006559275, −11.24375412292110, −10.93937571410992, −10.40246317712906, −10.19184573250455, −9.437482676226128, −9.017570557032616, −8.835680590331856, −8.176790274620823, −7.518245656987654, −7.309480714488083, −6.746129138289802, −6.301394880644145, −5.820990149702835, −5.379753411510532, −4.650175537663145, −4.428506509417202, −3.596974076831428, −3.077524429054674, −2.347211030530294, −2.149899992382677, −1.170494191241419, −0.6459803828982676, 0, 0.6459803828982676, 1.170494191241419, 2.149899992382677, 2.347211030530294, 3.077524429054674, 3.596974076831428, 4.428506509417202, 4.650175537663145, 5.379753411510532, 5.820990149702835, 6.301394880644145, 6.746129138289802, 7.309480714488083, 7.518245656987654, 8.176790274620823, 8.835680590331856, 9.017570557032616, 9.437482676226128, 10.19184573250455, 10.40246317712906, 10.93937571410992, 11.24375412292110, 11.57290006559275, 12.20916129536892, 12.77967423150328

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