Properties

Label 2-126-1.1-c3-0-4
Degree $2$
Conductor $126$
Sign $1$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s + 14·5-s − 7·7-s + 8·8-s + 28·10-s + 28·11-s + 18·13-s − 14·14-s + 16·16-s − 74·17-s + 80·19-s + 56·20-s + 56·22-s + 112·23-s + 71·25-s + 36·26-s − 28·28-s − 190·29-s + 72·31-s + 32·32-s − 148·34-s − 98·35-s − 346·37-s + 160·38-s + 112·40-s − 162·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.25·5-s − 0.377·7-s + 0.353·8-s + 0.885·10-s + 0.767·11-s + 0.384·13-s − 0.267·14-s + 1/4·16-s − 1.05·17-s + 0.965·19-s + 0.626·20-s + 0.542·22-s + 1.01·23-s + 0.567·25-s + 0.271·26-s − 0.188·28-s − 1.21·29-s + 0.417·31-s + 0.176·32-s − 0.746·34-s − 0.473·35-s − 1.53·37-s + 0.683·38-s + 0.442·40-s − 0.617·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.914875756\)
\(L(\frac12)\) \(\approx\) \(2.914875756\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 \)
7 \( 1 + p T \)
good5 \( 1 - 14 T + p^{3} T^{2} \)
11 \( 1 - 28 T + p^{3} T^{2} \)
13 \( 1 - 18 T + p^{3} T^{2} \)
17 \( 1 + 74 T + p^{3} T^{2} \)
19 \( 1 - 80 T + p^{3} T^{2} \)
23 \( 1 - 112 T + p^{3} T^{2} \)
29 \( 1 + 190 T + p^{3} T^{2} \)
31 \( 1 - 72 T + p^{3} T^{2} \)
37 \( 1 + 346 T + p^{3} T^{2} \)
41 \( 1 + 162 T + p^{3} T^{2} \)
43 \( 1 + 412 T + p^{3} T^{2} \)
47 \( 1 + 24 T + p^{3} T^{2} \)
53 \( 1 + 6 p T + p^{3} T^{2} \)
59 \( 1 - 200 T + p^{3} T^{2} \)
61 \( 1 + 198 T + p^{3} T^{2} \)
67 \( 1 + 716 T + p^{3} T^{2} \)
71 \( 1 + 392 T + p^{3} T^{2} \)
73 \( 1 - 538 T + p^{3} T^{2} \)
79 \( 1 - 240 T + p^{3} T^{2} \)
83 \( 1 - 1072 T + p^{3} T^{2} \)
89 \( 1 + 810 T + p^{3} T^{2} \)
97 \( 1 - 1354 T + p^{3} T^{2} \)
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

−13.28579591369027201251467138572, −12.00270826923332995709534680239, −10.93188645914146430158975709895, −9.770377891637989890059260042903, −8.869144705283915979044779159613, −7.01463058785360649024309836730, −6.14339495874195130597288620560, −5.01848454620816188995181327211, −3.36798678013953921936490236760, −1.74512713496138990197795606475, 1.74512713496138990197795606475, 3.36798678013953921936490236760, 5.01848454620816188995181327211, 6.14339495874195130597288620560, 7.01463058785360649024309836730, 8.869144705283915979044779159613, 9.770377891637989890059260042903, 10.93188645914146430158975709895, 12.00270826923332995709534680239, 13.28579591369027201251467138572

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