Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 31^{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 + 11-s − 12-s + 4·13-s − 2·14-s + 16-s + 6·17-s − 18-s − 4·19-s − 2·21-s − 22-s − 6·23-s + 24-s − 5·25-s − 4·26-s − 27-s + 2·28-s − 6·29-s − 32-s − 33-s − 6·34-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 + 0.301·11-s − 0.288·12-s + 1.10·13-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 − 0.213·22-s − 1.25·23-s + 0.204·24-s − 25-s − 0.784·26-s − 0.192·27-s + 0.377·28-s − 1.11·29-s − 0.176·32-s − 0.174·33-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63426 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63426 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(63426\)    =    \(2 \cdot 3 \cdot 11 \cdot 31^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{63426} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 63426,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;31\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
11 \( 1 - T \)
31 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
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

−14.63049693876854, −14.11866631939155, −13.38545417182462, −12.98323841021487, −12.30834002422398, −11.78714802872421, −11.41665779346423, −11.02681838100025, −10.43408433575564, −9.921630425386471, −9.500302309807247, −8.846028173346778, −8.181937000261413, −7.829101205682158, −7.503116806946150, −6.525302254082838, −6.082891502893094, −5.795491597278805, −5.039392460860705, −4.260843669064462, −3.800674318849691, −3.105416592374674, −2.018922290816023, −1.639242748693367, −0.9302797465223217, 0, 0.9302797465223217, 1.639242748693367, 2.018922290816023, 3.105416592374674, 3.800674318849691, 4.260843669064462, 5.039392460860705, 5.795491597278805, 6.082891502893094, 6.525302254082838, 7.503116806946150, 7.829101205682158, 8.181937000261413, 8.846028173346778, 9.500302309807247, 9.921630425386471, 10.43408433575564, 11.02681838100025, 11.41665779346423, 11.78714802872421, 12.30834002422398, 12.98323841021487, 13.38545417182462, 14.11866631939155, 14.63049693876854

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