↑

Properties

Label 2-139650-1.1-c1-0-121
Degree $2$
Conductor $139650$
Sign $-1$
Analytic cond. $1115.11$
Root an. cond. $33.3932$
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 + 8-s + 9-s − 6·11-s − 12-s + 2·13-s + 16-s − 4·17-s + 18-s − 19-s − 6·22-s − 24-s + 2·26-s − 27-s − 8·29-s − 2·31-s + 32-s + 6·33-s − 4·34-s + 36-s − 38-s − 2·39-s − 2·41-s + 6·43-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s + 0.554·13-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.229·19-s − 1.27·22-s − 0.204·24-s + 0.392·26-s − 0.192·27-s − 1.48·29-s − 0.359·31-s + 0.176·32-s + 1.04·33-s − 0.685·34-s + 1/6·36-s − 0.162·38-s − 0.320·39-s − 0.312·41-s + 0.914·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(139650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(1115.11\)
Root analytic conductor: \(33.3932\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 + T \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−13.44402992840694, −13.22713197231697, −12.72266556741794, −12.39599720155803, −11.74989349572285, −11.09572938327297, −10.92870729624925, −10.56238733045862, −9.955893097407258, −9.357796340182860, −8.754294110026301, −8.211698554459975, −7.606023574860652, −7.248976868695665, −6.703255033550072, −5.949035298012801, −5.743839843655091, −5.153837738915721, −4.717415659496596, −4.065522033092749, −3.604143482928762, −2.821071720210834, −2.269402181291200, −1.783513013565452, −0.7403531705691297, 0, 0.7403531705691297, 1.783513013565452, 2.269402181291200, 2.821071720210834, 3.604143482928762, 4.065522033092749, 4.717415659496596, 5.153837738915721, 5.743839843655091, 5.949035298012801, 6.703255033550072, 7.248976868695665, 7.606023574860652, 8.211698554459975, 8.754294110026301, 9.357796340182860, 9.955893097407258, 10.56238733045862, 10.92870729624925, 11.09572938327297, 11.74989349572285, 12.39599720155803, 12.72266556741794, 13.22713197231697, 13.44402992840694

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