Properties

Degree 2
Conductor $ 2 \cdot 7^{2} \cdot 17^{2} $
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-s + 4-s + 8-s − 3·9-s − 2·13-s + 16-s − 3·18-s + 4·19-s + 8·23-s − 5·25-s − 2·26-s + 8·29-s − 8·31-s + 32-s − 3·36-s − 8·37-s + 4·38-s − 8·41-s + 4·43-s + 8·46-s + 8·47-s − 5·50-s − 2·52-s + 10·53-s + 8·58-s − 12·59-s − 8·62-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s − 0.554·13-s + 1/4·16-s − 0.707·18-s + 0.917·19-s + 1.66·23-s − 25-s − 0.392·26-s + 1.48·29-s − 1.43·31-s + 0.176·32-s − 1/2·36-s − 1.31·37-s + 0.648·38-s − 1.24·41-s + 0.609·43-s + 1.17·46-s + 1.16·47-s − 0.707·50-s − 0.277·52-s + 1.37·53-s + 1.05·58-s − 1.56·59-s − 1.01·62-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  =  \(1\)
Selberg data  =  \((2,\ 28322,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

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

−15.34612951562099, −14.95022389949213, −14.31737296578856, −13.83730917659387, −13.56782796564482, −12.79443935166205, −12.21567495290406, −11.81722411405080, −11.38311690044992, −10.58206969520784, −10.36779937247235, −9.382565611488252, −8.981235730770135, −8.381963964829852, −7.578503456065338, −7.156664786362052, −6.578629607559805, −5.694863928046418, −5.417098874543795, −4.819897857221045, −4.057635762065836, −3.213486521335796, −2.907797167076729, −2.057944769045559, −1.138957943856753, 0, 1.138957943856753, 2.057944769045559, 2.907797167076729, 3.213486521335796, 4.057635762065836, 4.819897857221045, 5.417098874543795, 5.694863928046418, 6.578629607559805, 7.156664786362052, 7.578503456065338, 8.381963964829852, 8.981235730770135, 9.382565611488252, 10.36779937247235, 10.58206969520784, 11.38311690044992, 11.81722411405080, 12.21567495290406, 12.79443935166205, 13.56782796564482, 13.83730917659387, 14.31737296578856, 14.95022389949213, 15.34612951562099

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