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

−15.38606192451109, −14.75324843646305, −14.42832607986249, −14.10704695849659, −13.37891269910040, −12.85548639745344, −12.20250281765856, −11.71807226493302, −11.50802974118007, −10.81140685839452, −9.949375308535382, −9.608979050023096, −8.809403860313312, −8.233429397395277, −7.791495548140976, −7.228299630016944, −6.583032348318255, −5.952259845522505, −5.232484651066927, −4.553929963119601, −4.061232217147231, −3.378982141041921, −2.763895238806491, −2.237746343848418, −1.106272208052234, 0, 1.106272208052234, 2.237746343848418, 2.763895238806491, 3.378982141041921, 4.061232217147231, 4.553929963119601, 5.232484651066927, 5.952259845522505, 6.583032348318255, 7.228299630016944, 7.791495548140976, 8.233429397395277, 8.809403860313312, 9.608979050023096, 9.949375308535382, 10.81140685839452, 11.50802974118007, 11.71807226493302, 12.20250281765856, 12.85548639745344, 13.37891269910040, 14.10704695849659, 14.42832607986249, 14.75324843646305, 15.38606192451109

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