Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11^{2} \cdot 13^{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 − 6-s + 2·7-s − 8-s + 9-s + 12-s − 2·14-s + 16-s + 6·17-s − 18-s − 4·19-s + 2·21-s + 6·23-s − 24-s − 5·25-s + 27-s + 2·28-s − 6·29-s − 8·31-s − 32-s − 6·34-s + 36-s + 10·37-s + 4·38-s + 6·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.436·21-s + 1.25·23-s − 0.204·24-s − 25-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 1.64·37-s + 0.648·38-s + 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(122694\)    =    \(2 \cdot 3 \cdot 11^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{122694} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 122694,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−13.85898641581309, −13.24249273793869, −12.75841602867172, −12.45447023786586, −11.66528042486552, −11.24887471808842, −10.94758244978017, −10.31177549855240, −9.768450670856516, −9.394858683513019, −8.893850617172715, −8.375984866767383, −7.885094049935532, −7.475387524234915, −7.165348729192886, −6.352879836233499, −5.687882756897469, −5.417162844553117, −4.512560163900048, −4.068838670266899, −3.339864826155550, −2.853823649248542, −2.065157489066691, −1.607272804470768, −0.9799798409498989, 0, 0.9799798409498989, 1.607272804470768, 2.065157489066691, 2.853823649248542, 3.339864826155550, 4.068838670266899, 4.512560163900048, 5.417162844553117, 5.687882756897469, 6.352879836233499, 7.165348729192886, 7.475387524234915, 7.885094049935532, 8.375984866767383, 8.893850617172715, 9.394858683513019, 9.768450670856516, 10.31177549855240, 10.94758244978017, 11.24887471808842, 11.66528042486552, 12.45447023786586, 12.75841602867172, 13.24249273793869, 13.85898641581309

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