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 + 3·3-s + 4-s − 5-s − 3·6-s + 2·7-s − 8-s + 6·9-s + 10-s − 4·11-s + 3·12-s − 3·13-s − 2·14-s − 3·15-s + 16-s + 17-s − 6·18-s + 3·19-s − 20-s + 6·21-s + 4·22-s − 6·23-s − 3·24-s + 25-s + 3·26-s + 9·27-s + 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.22·6-s + 0.755·7-s − 0.353·8-s + 2·9-s + 0.316·10-s − 1.20·11-s + 0.866·12-s − 0.832·13-s − 0.534·14-s − 0.774·15-s + 1/4·16-s + 0.242·17-s − 1.41·18-s + 0.688·19-s − 0.223·20-s + 1.30·21-s + 0.852·22-s − 1.25·23-s − 0.612·24-s + 1/5·25-s + 0.588·26-s + 1.73·27-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

\( d \)  =  \(2\)
\( N \)  =  \(170\)    =    \(2 \cdot 5 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{170} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 170,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.336025674$
$L(\frac12)$  $\approx$  $1.336025674$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−19.75278221228320, −19.29023885624343, −18.31272083418862, −17.67472326570416, −16.09094545106518, −15.60842280149877, −14.58562176095815, −14.03660728517763, −12.83373639226607, −11.77392021354054, −10.36366338935226, −9.673175371223345, −8.338175486309026, −8.047148785250350, −7.122273982893580, −4.920150647777939, −3.294603148581971, −2.088810387374048, 2.088810387374048, 3.294603148581971, 4.920150647777939, 7.122273982893580, 8.047148785250350, 8.338175486309026, 9.673175371223345, 10.36366338935226, 11.77392021354054, 12.83373639226607, 14.03660728517763, 14.58562176095815, 15.60842280149877, 16.09094545106518, 17.67472326570416, 18.31272083418862, 19.29023885624343, 19.75278221228320

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