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 + 4·5-s − 2·6-s − 8-s + 9-s − 4·10-s + 4·11-s + 2·12-s + 4·13-s + 8·15-s + 16-s − 18-s + 6·19-s + 4·20-s − 4·22-s − 2·24-s + 11·25-s − 4·26-s − 4·27-s − 6·29-s − 8·30-s + 4·31-s − 32-s + 8·33-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s + 1.78·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s − 1.26·10-s + 1.20·11-s + 0.577·12-s + 1.10·13-s + 2.06·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s + 0.894·20-s − 0.852·22-s − 0.408·24-s + 11/5·25-s − 0.784·26-s − 0.769·27-s − 1.11·29-s − 1.46·30-s + 0.718·31-s − 0.176·32-s + 1.39·33-s + 1/6·36-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  =  \((2,\ 28322,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(5.283537652\)
\(L(\frac12)\)  \(\approx\)  \(5.283537652\)
\(L(\frac{3}{2})\)   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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.97890794938574, −14.67215496525757, −14.04769512040545, −13.76084878765563, −13.28737718026147, −12.77957032975708, −11.87948865863843, −11.31544702381864, −10.84402187294521, −9.889205653721615, −9.729793138782635, −9.180477838724779, −8.894780525980430, −8.269735484832030, −7.607454591978951, −6.936271475816535, −6.263339492616406, −5.860701507171603, −5.281931253482242, −4.129296764322135, −3.523328792577930, −2.728130871018458, −2.291985042725067, −1.375029723760508, −1.099881909436543, 1.099881909436543, 1.375029723760508, 2.291985042725067, 2.728130871018458, 3.523328792577930, 4.129296764322135, 5.281931253482242, 5.860701507171603, 6.263339492616406, 6.936271475816535, 7.607454591978951, 8.269735484832030, 8.894780525980430, 9.180477838724779, 9.729793138782635, 9.889205653721615, 10.84402187294521, 11.31544702381864, 11.87948865863843, 12.77957032975708, 13.28737718026147, 13.76084878765563, 14.04769512040545, 14.67215496525757, 14.97890794938574

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