Properties

Label 2-126-1.1-c1-0-0
Degree $2$
Conductor $126$
Sign $1$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·5-s − 7-s − 8-s − 2·10-s + 4·11-s + 6·13-s + 14-s + 16-s − 2·17-s − 4·19-s + 2·20-s − 4·22-s − 8·23-s − 25-s − 6·26-s − 28-s + 2·29-s − 32-s + 2·34-s − 2·35-s − 10·37-s + 4·38-s − 2·40-s + 6·41-s − 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 0.632·10-s + 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.447·20-s − 0.852·22-s − 1.66·23-s − 1/5·25-s − 1.17·26-s − 0.188·28-s + 0.371·29-s − 0.176·32-s + 0.342·34-s − 0.338·35-s − 1.64·37-s + 0.648·38-s − 0.316·40-s + 0.937·41-s − 0.609·43-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{126} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9104737368\)
\(L(\frac12)\) \(\approx\) \(0.9104737368\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.48577504303344657036010789975, −12.25720048996309464371398882337, −11.12015185557948517225700432181, −10.13805865622749455803240285872, −9.164662289745378790951914691450, −8.337652051623380123011624806396, −6.58813712157544804919343144385, −6.01899647743800981343957937964, −3.83863171110134241641513411613, −1.79365373540820646322667624458, 1.79365373540820646322667624458, 3.83863171110134241641513411613, 6.01899647743800981343957937964, 6.58813712157544804919343144385, 8.337652051623380123011624806396, 9.164662289745378790951914691450, 10.13805865622749455803240285872, 11.12015185557948517225700432181, 12.25720048996309464371398882337, 13.48577504303344657036010789975

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