Properties

Degree 2
Conductor $ 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 + 2·3-s − 4-s − 5-s + 2·6-s − 2·7-s − 3·8-s + 9-s − 10-s + 2·11-s − 2·12-s + 2·13-s − 2·14-s − 2·15-s − 16-s + 17-s + 18-s + 20-s − 4·21-s + 2·22-s + 6·23-s − 6·24-s + 25-s + 2·26-s − 4·27-s + 2·28-s − 6·29-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 − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-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 + 0.223·20-s − 0.872·21-s + 0.426·22-s + 1.25·23-s − 1.22·24-s + 1/5·25-s + 0.392·26-s − 0.769·27-s + 0.377·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(85\)    =    \(5 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{85} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 85,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.397692218$
$L(\frac12)$  $\approx$  $1.397692218$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + T \)
17 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.20287490779547, −18.60137008410991, −17.17323462124726, −15.93519362714049, −14.85386522256541, −14.32499099465890, −13.23869509800645, −12.61695801969404, −11.16926979542543, −9.342260097121774, −8.918128413786971, −7.466130876438910, −5.833982579727286, −4.032977799112472, −3.137575258423249, 3.137575258423249, 4.032977799112472, 5.833982579727286, 7.466130876438910, 8.918128413786971, 9.342260097121774, 11.16926979542543, 12.61695801969404, 13.23869509800645, 14.32499099465890, 14.85386522256541, 15.93519362714049, 17.17323462124726, 18.60137008410991, 19.20287490779547

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