Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 7^{2} \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 9-s + 11-s + 2·13-s + 2·15-s − 4·17-s + 6·19-s − 25-s − 27-s − 8·29-s + 8·31-s − 33-s + 10·37-s − 2·39-s − 8·41-s − 2·43-s − 2·45-s + 8·47-s + 4·51-s − 2·53-s − 2·55-s − 6·57-s − 12·59-s − 10·61-s − 4·65-s + 12·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.516·15-s − 0.970·17-s + 1.37·19-s − 1/5·25-s − 0.192·27-s − 1.48·29-s + 1.43·31-s − 0.174·33-s + 1.64·37-s − 0.320·39-s − 1.24·41-s − 0.304·43-s − 0.298·45-s + 1.16·47-s + 0.560·51-s − 0.274·53-s − 0.269·55-s − 0.794·57-s − 1.56·59-s − 1.28·61-s − 0.496·65-s + 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6468\)    =    \(2^{2} \cdot 3 \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6468} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6468,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.149102567$
$L(\frac12)$  $\approx$  $1.149102567$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−17.09962182168900, −16.88299445108762, −15.98832858655378, −15.57182970452072, −15.26058268557606, −14.34086278148564, −13.61410287719079, −13.19255815204438, −12.34068422227135, −11.75874395258221, −11.35891363794526, −10.88374449498768, −9.980201857752060, −9.374776077539456, −8.668894887992180, −7.822215382272755, −7.428461241601778, −6.542972868096665, −5.989408844311772, −5.116715878604552, −4.361673571374373, −3.758225789977090, −2.890086874470534, −1.642529003441522, −0.5871314610938338, 0.5871314610938338, 1.642529003441522, 2.890086874470534, 3.758225789977090, 4.361673571374373, 5.116715878604552, 5.989408844311772, 6.542972868096665, 7.428461241601778, 7.822215382272755, 8.668894887992180, 9.374776077539456, 9.980201857752060, 10.88374449498768, 11.35891363794526, 11.75874395258221, 12.34068422227135, 13.19255815204438, 13.61410287719079, 14.34086278148564, 15.26058268557606, 15.57182970452072, 15.98832858655378, 16.88299445108762, 17.09962182168900

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