Properties

Degree 2
Conductor $ 2^{2} \cdot 43 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·7-s + 9-s − 3·11-s − 13-s − 3·17-s + 2·19-s + 8·21-s − 3·23-s − 5·25-s + 4·27-s + 6·29-s + 5·31-s + 6·33-s + 8·37-s + 2·39-s − 3·41-s + 43-s − 12·47-s + 9·49-s + 6·51-s − 9·53-s − 4·57-s − 12·59-s − 10·61-s − 4·63-s + 11·67-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.51·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s − 0.727·17-s + 0.458·19-s + 1.74·21-s − 0.625·23-s − 25-s + 0.769·27-s + 1.11·29-s + 0.898·31-s + 1.04·33-s + 1.31·37-s + 0.320·39-s − 0.468·41-s + 0.152·43-s − 1.75·47-s + 9/7·49-s + 0.840·51-s − 1.23·53-s − 0.529·57-s − 1.56·59-s − 1.28·61-s − 0.503·63-s + 1.34·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 172 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 172 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(172\)    =    \(2^{2} \cdot 43\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{172} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 172,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;43\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;43\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
43 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−19.50060128190495, −18.42015557697680, −17.71434993341013, −16.82157728838981, −15.97835842701822, −15.55191222698312, −13.93094482524760, −13.02374166175235, −12.24489403783955, −11.32440648392506, −10.25845810315302, −9.566827060207866, −8.016339133906283, −6.605976953240298, −5.970576940713823, −4.691087978287624, −2.939534926582883, 0, 2.939534926582883, 4.691087978287624, 5.970576940713823, 6.605976953240298, 8.016339133906283, 9.566827060207866, 10.25845810315302, 11.32440648392506, 12.24489403783955, 13.02374166175235, 13.93094482524760, 15.55191222698312, 15.97835842701822, 16.82157728838981, 17.71434993341013, 18.42015557697680, 19.50060128190495

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