Properties

Label 2-12696-1.1-c1-0-10
Degree $2$
Conductor $12696$
Sign $-1$
Analytic cond. $101.378$
Root an. cond. $10.0686$
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  − 3-s − 2·5-s − 2·7-s + 9-s + 2·13-s + 2·15-s + 6·17-s − 2·19-s + 2·21-s − 25-s − 27-s − 6·29-s + 4·35-s − 4·37-s − 2·39-s + 2·41-s + 6·43-s − 2·45-s − 4·47-s − 3·49-s − 6·51-s − 6·53-s + 2·57-s + 4·61-s − 2·63-s − 4·65-s + 10·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.458·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.676·35-s − 0.657·37-s − 0.320·39-s + 0.312·41-s + 0.914·43-s − 0.298·45-s − 0.583·47-s − 3/7·49-s − 0.840·51-s − 0.824·53-s + 0.264·57-s + 0.512·61-s − 0.251·63-s − 0.496·65-s + 1.22·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12696\)    =    \(2^{3} \cdot 3 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(101.378\)
Root analytic conductor: \(10.0686\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{12696} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 12696,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
23 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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

−16.50829079615868, −16.00971951466519, −15.64491251917748, −14.97803508455140, −14.38742070668832, −13.74234457851948, −12.86863455317934, −12.69943866362257, −11.95912216811279, −11.52101690665867, −10.87388129752083, −10.36967761645660, −9.599622053288786, −9.204499211785280, −8.143044928153716, −7.870448816550289, −7.082662623828338, −6.480110982754820, −5.800436704659364, −5.249569721727065, −4.318159441120237, −3.638210032477148, −3.240512148514403, −2.010749277057079, −0.9336902404482082, 0, 0.9336902404482082, 2.010749277057079, 3.240512148514403, 3.638210032477148, 4.318159441120237, 5.249569721727065, 5.800436704659364, 6.480110982754820, 7.082662623828338, 7.870448816550289, 8.143044928153716, 9.204499211785280, 9.599622053288786, 10.36967761645660, 10.87388129752083, 11.52101690665867, 11.95912216811279, 12.69943866362257, 12.86863455317934, 13.74234457851948, 14.38742070668832, 14.97803508455140, 15.64491251917748, 16.00971951466519, 16.50829079615868

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