Properties

Label 2-12696-1.1-c1-0-7
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s + 4·7-s + 9-s + 2·11-s + 6·13-s − 2·15-s − 6·19-s + 4·21-s − 25-s + 27-s + 6·29-s − 4·31-s + 2·33-s − 8·35-s − 2·37-s + 6·39-s + 6·41-s − 2·43-s − 2·45-s + 4·47-s + 9·49-s + 14·53-s − 4·55-s − 6·57-s − 4·59-s + 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.603·11-s + 1.66·13-s − 0.516·15-s − 1.37·19-s + 0.872·21-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.348·33-s − 1.35·35-s − 0.328·37-s + 0.960·39-s + 0.937·41-s − 0.304·43-s − 0.298·45-s + 0.583·47-s + 9/7·49-s + 1.92·53-s − 0.539·55-s − 0.794·57-s − 0.520·59-s + 1.28·61-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: \(0\)
Selberg data: \((2,\ 12696,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.265169718\)
\(L(\frac12)\) \(\approx\) \(3.265169718\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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

−16.08819604036670, −15.67914234096790, −15.09951936077731, −14.64557842272568, −14.12602708537238, −13.61155888949480, −12.92758300452340, −12.23513340388058, −11.64980932036043, −11.12569532105665, −10.76765950628994, −10.02924118819919, −8.964039069229445, −8.543254972697866, −8.309883914739435, −7.603022479050704, −6.945372148697959, −6.192303800089676, −5.432600951333801, −4.461035850676079, −4.071256937359413, −3.566057001939091, −2.415762663294877, −1.643818872988273, −0.8485740766815657, 0.8485740766815657, 1.643818872988273, 2.415762663294877, 3.566057001939091, 4.071256937359413, 4.461035850676079, 5.432600951333801, 6.192303800089676, 6.945372148697959, 7.603022479050704, 8.309883914739435, 8.543254972697866, 8.964039069229445, 10.02924118819919, 10.76765950628994, 11.12569532105665, 11.64980932036043, 12.23513340388058, 12.92758300452340, 13.61155888949480, 14.12602708537238, 14.64557842272568, 15.09951936077731, 15.67914234096790, 16.08819604036670

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