Properties

Label 2-12696-1.1-c1-0-6
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

\(L(1)\) \(\approx\) \(2.940263447\)
\(L(\frac12)\) \(\approx\) \(2.940263447\)
\(L(\frac{3}{2})\) 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

−16.07708833315165, −15.76974964234109, −15.48108896950152, −14.27879348179978, −14.03207639802217, −13.35062088637005, −13.16523531937288, −12.53237819060399, −11.84239279828164, −10.88703338608986, −10.50497875937369, −9.786440662479038, −9.299585969991073, −9.020004534305415, −8.040630997045963, −7.551383107379555, −6.635628272513257, −6.130137536385021, −5.730060859195071, −4.799489962781379, −3.784634680511370, −3.227836773185010, −2.701256671064908, −1.698529528987115, −0.7638876467743345, 0.7638876467743345, 1.698529528987115, 2.701256671064908, 3.227836773185010, 3.784634680511370, 4.799489962781379, 5.730060859195071, 6.130137536385021, 6.635628272513257, 7.551383107379555, 8.040630997045963, 9.020004534305415, 9.299585969991073, 9.786440662479038, 10.50497875937369, 10.88703338608986, 11.84239279828164, 12.53237819060399, 13.16523531937288, 13.35062088637005, 14.03207639802217, 14.27879348179978, 15.48108896950152, 15.76974964234109, 16.07708833315165

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