Properties

Degree $2$
Conductor $170$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s − 5-s + 2·6-s + 2·7-s − 8-s + 9-s + 10-s + 6·11-s − 2·12-s + 2·13-s − 2·14-s + 2·15-s + 16-s + 17-s − 18-s + 8·19-s − 20-s − 4·21-s − 6·22-s − 6·23-s + 2·24-s + 25-s − 2·26-s + 4·27-s + 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.577·12-s + 0.554·13-s − 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.83·19-s − 0.223·20-s − 0.872·21-s − 1.27·22-s − 1.25·23-s + 0.408·24-s + 1/5·25-s − 0.392·26-s + 0.769·27-s + 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{170} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6372606581\)
\(L(\frac12)\) \(\approx\) \(0.6372606581\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + T \)
17 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−18.79840124407033, −18.05360344479791, −17.34428825688615, −16.63665084160105, −15.93653843517822, −14.74712366362145, −13.84903441790987, −11.98273905819748, −11.79012980016461, −11.03367638212706, −9.816164727688897, −8.727177005854591, −7.568841180917286, −6.455853923374890, −5.405897167600434, −3.840993542941173, −1.228092592679110, 1.228092592679110, 3.840993542941173, 5.405897167600434, 6.455853923374890, 7.568841180917286, 8.727177005854591, 9.816164727688897, 11.03367638212706, 11.79012980016461, 11.98273905819748, 13.84903441790987, 14.74712366362145, 15.93653843517822, 16.63665084160105, 17.34428825688615, 18.05360344479791, 18.79840124407033

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