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 − 3-s + 4-s + 2·5-s − 6-s + 8-s − 2·9-s + 2·10-s + 2·11-s − 12-s + 4·13-s − 2·15-s + 16-s − 2·18-s − 19-s + 2·20-s + 2·22-s − 24-s − 25-s + 4·26-s + 5·27-s + 3·29-s − 2·30-s − 7·31-s + 32-s − 2·33-s − 2·36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.632·10-s + 0.603·11-s − 0.288·12-s + 1.10·13-s − 0.516·15-s + 1/4·16-s − 0.471·18-s − 0.229·19-s + 0.447·20-s + 0.426·22-s − 0.204·24-s − 1/5·25-s + 0.784·26-s + 0.962·27-s + 0.557·29-s − 0.365·30-s − 1.25·31-s + 0.176·32-s − 0.348·33-s − 1/3·36-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  =  \(1\)
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 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−15.49679610738810, −14.79338781045994, −14.26484985179863, −13.94133267805013, −13.32669147876151, −12.97816220400699, −12.16537518597137, −11.84872409147005, −11.19188300198797, −10.76554523763721, −10.30730136163824, −9.464566085113266, −9.034226468215274, −8.370763535706658, −7.745840342963355, −6.726726984356045, −6.515670648486205, −5.732237540785190, −5.650673911736354, −4.780594552189101, −4.132120720313332, −3.360641852412749, −2.791912716639016, −1.788294863235343, −1.313489896361748, 0, 1.313489896361748, 1.788294863235343, 2.791912716639016, 3.360641852412749, 4.132120720313332, 4.780594552189101, 5.650673911736354, 5.732237540785190, 6.515670648486205, 6.726726984356045, 7.745840342963355, 8.370763535706658, 9.034226468215274, 9.464566085113266, 10.30730136163824, 10.76554523763721, 11.19188300198797, 11.84872409147005, 12.16537518597137, 12.97816220400699, 13.32669147876151, 13.94133267805013, 14.26484985179863, 14.79338781045994, 15.49679610738810

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