Properties

Label 2-17136-1.1-c1-0-42
Degree $2$
Conductor $17136$
Sign $-1$
Analytic cond. $136.831$
Root an. cond. $11.6975$
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  + 2·5-s + 7-s + 2·13-s − 17-s − 25-s − 2·29-s − 8·31-s + 2·35-s − 6·37-s + 6·41-s − 4·43-s + 49-s − 14·53-s − 8·59-s + 14·61-s + 4·65-s − 4·67-s + 8·71-s + 10·73-s − 8·79-s − 16·83-s − 2·85-s − 2·89-s + 2·91-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s + 0.554·13-s − 0.242·17-s − 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.338·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s − 1.92·53-s − 1.04·59-s + 1.79·61-s + 0.496·65-s − 0.488·67-s + 0.949·71-s + 1.17·73-s − 0.900·79-s − 1.75·83-s − 0.216·85-s − 0.211·89-s + 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17136\)    =    \(2^{4} \cdot 3^{2} \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(136.831\)
Root analytic conductor: \(11.6975\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{17136} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 17136,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
17 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−15.96434206210129, −15.78562335943200, −14.92800494691699, −14.43414660958547, −13.94221862625242, −13.46807895133150, −12.81575629125638, −12.45992249149900, −11.54191852711651, −11.09327781296386, −10.62300982068259, −9.871269109607739, −9.395991667016805, −8.856480911871278, −8.180157365909502, −7.581892788578717, −6.810403660099002, −6.281673330038136, −5.519767247309473, −5.201584799011292, −4.228750219776935, −3.613720792946470, −2.718930582510086, −1.880175919234128, −1.373284341172561, 0, 1.373284341172561, 1.880175919234128, 2.718930582510086, 3.613720792946470, 4.228750219776935, 5.201584799011292, 5.519767247309473, 6.281673330038136, 6.810403660099002, 7.581892788578717, 8.180157365909502, 8.856480911871278, 9.395991667016805, 9.871269109607739, 10.62300982068259, 11.09327781296386, 11.54191852711651, 12.45992249149900, 12.81575629125638, 13.46807895133150, 13.94221862625242, 14.43414660958547, 14.92800494691699, 15.78562335943200, 15.96434206210129

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