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 − 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 − 26-s − 5·27-s + 2·28-s − 3·29-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 − 0.277·13-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 − 0.196·26-s − 0.962·27-s + 0.377·28-s − 0.557·29-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  =  $(2,\ 170,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.846901354$
$L(\frac12)$  $\approx$  $1.846901354$
$L(\frac{3}{2})$   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 \( 1 + T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
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

−19.40491720503786, −18.22814374492658, −17.25143186934982, −16.27171306849706, −15.22943981290743, −14.57638717555755, −13.90322550126828, −12.86758764551412, −11.73207427636390, −11.16705478363741, −9.780585800377851, −8.390318499248391, −7.756988657177374, −6.296659908964156, −4.976399639835590, −3.775011686616967, −2.330459292325687, 2.330459292325687, 3.775011686616967, 4.976399639835590, 6.296659908964156, 7.756988657177374, 8.390318499248391, 9.780585800377851, 11.16705478363741, 11.73207427636390, 12.86758764551412, 13.90322550126828, 14.57638717555755, 15.22943981290743, 16.27171306849706, 17.25143186934982, 18.22814374492658, 19.40491720503786

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