Properties

Degree 2
Conductor $ 2 \cdot 7^{2} \cdot 17^{2} $
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 + 2·6-s − 8-s + 9-s − 2·12-s + 4·13-s + 16-s − 18-s − 2·19-s + 2·24-s − 5·25-s − 4·26-s + 4·27-s + 6·29-s − 4·31-s − 32-s + 36-s − 2·37-s + 2·38-s − 8·39-s + 6·41-s + 8·43-s + 12·47-s − 2·48-s + 5·50-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.577·12-s + 1.10·13-s + 1/4·16-s − 0.235·18-s − 0.458·19-s + 0.408·24-s − 25-s − 0.784·26-s + 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 1/6·36-s − 0.328·37-s + 0.324·38-s − 1.28·39-s + 0.937·41-s + 1.21·43-s + 1.75·47-s − 0.288·48-s + 0.707·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(28322\)    =    \(2 \cdot 7^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{28322} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 28322,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.9720606617\)
\(L(\frac12)\)  \(\approx\)  \(0.9720606617\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

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

−15.52438442525064, −14.73656500758754, −14.19557060041075, −13.54484083780491, −12.99404196784882, −12.16852999619247, −12.06933226531708, −11.24977310470779, −10.93024025151654, −10.48927574534232, −9.944712601885258, −9.139429875069651, −8.727997990211820, −8.125592459516536, −7.451920801474087, −6.804542028509099, −6.279489933602623, −5.700185020196356, −5.390015840914639, −4.295396462186042, −3.886972406084437, −2.849335745149190, −2.114873981625971, −1.125790132487681, −0.5395601724337955, 0.5395601724337955, 1.125790132487681, 2.114873981625971, 2.849335745149190, 3.886972406084437, 4.295396462186042, 5.390015840914639, 5.700185020196356, 6.279489933602623, 6.804542028509099, 7.451920801474087, 8.125592459516536, 8.727997990211820, 9.139429875069651, 9.944712601885258, 10.48927574534232, 10.93024025151654, 11.24977310470779, 12.06933226531708, 12.16852999619247, 12.99404196784882, 13.54484083780491, 14.19557060041075, 14.73656500758754, 15.52438442525064

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