Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 17 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 2·7-s − 8-s − 2·9-s − 10-s + 12-s + 5·13-s − 2·14-s + 15-s + 16-s − 17-s + 2·18-s − 19-s + 20-s + 2·21-s + 6·23-s − 24-s + 25-s − 5·26-s − 5·27-s + 2·28-s − 9·29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.288·12-s + 1.38·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.471·18-s − 0.229·19-s + 0.223·20-s + 0.436·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s − 0.962·27-s + 0.377·28-s − 1.67·29-s − 0.182·30-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

\( d \)  =  \(2\)
\( N \)  =  \(170\)    =    \(2 \cdot 5 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{170} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 170,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.109702059$
$L(\frac12)$  $\approx$  $1.109702059$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
5 \( 1 - T \)
17 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.95126638742626, −18.94257998913926, −18.25665978688176, −17.35939737869670, −16.69605922076646, −15.48028934591048, −14.70421880778465, −13.80856459823240, −12.88999568022307, −11.31002289164134, −10.94841428782721, −9.477504028752913, −8.698484614188418, −7.958366536731622, −6.573129040749490, −5.321452082421392, −3.403443709451677, −1.795790547589078, 1.795790547589078, 3.403443709451677, 5.321452082421392, 6.573129040749490, 7.958366536731622, 8.698484614188418, 9.477504028752913, 10.94841428782721, 11.31002289164134, 12.88999568022307, 13.80856459823240, 14.70421880778465, 15.48028934591048, 16.69605922076646, 17.35939737869670, 18.25665978688176, 18.94257998913926, 19.95126638742626

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