Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7^{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  + 3-s − 2·5-s + 9-s − 4·11-s − 2·13-s − 2·15-s + 6·17-s + 4·19-s − 25-s + 27-s + 2·29-s − 4·33-s − 6·37-s − 2·39-s − 2·41-s + 4·43-s − 2·45-s + 6·51-s − 6·53-s + 8·55-s + 4·57-s + 12·59-s − 2·61-s + 4·65-s − 4·67-s + 6·73-s − 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.696·33-s − 0.986·37-s − 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.298·45-s + 0.840·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s + 1.56·59-s − 0.256·61-s + 0.496·65-s − 0.488·67-s + 0.702·73-s − 0.115·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{9408} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 9408,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−16.94893313330074, −16.21806523977073, −15.79118541937749, −15.45009322273932, −14.64334100782154, −14.25745228461106, −13.60150147680276, −12.94555669779658, −12.30455745477888, −11.91154810945705, −11.21722475929760, −10.38669416399315, −9.979002648294755, −9.360697556527118, −8.400771206570613, −8.047316704349056, −7.411629795633274, −7.084632846144490, −5.843666831694476, −5.252330618050711, −4.567787127119415, −3.634173949119671, −3.132954490048871, −2.368228254724345, −1.196512422822153, 0, 1.196512422822153, 2.368228254724345, 3.132954490048871, 3.634173949119671, 4.567787127119415, 5.252330618050711, 5.843666831694476, 7.084632846144490, 7.411629795633274, 8.047316704349056, 8.400771206570613, 9.360697556527118, 9.979002648294755, 10.38669416399315, 11.21722475929760, 11.91154810945705, 12.30455745477888, 12.94555669779658, 13.60150147680276, 14.25745228461106, 14.64334100782154, 15.45009322273932, 15.79118541937749, 16.21806523977073, 16.94893313330074

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