Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 11 \cdot 19^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·7-s − 8-s + 11-s + 4·13-s − 2·14-s + 16-s + 6·17-s − 22-s − 6·23-s − 5·25-s − 4·26-s + 2·28-s + 6·29-s − 8·31-s − 32-s − 6·34-s + 10·37-s + 6·41-s + 8·43-s + 44-s + 6·46-s + 6·47-s − 3·49-s + 5·50-s + 4·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.301·11-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.213·22-s − 1.25·23-s − 25-s − 0.784·26-s + 0.377·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 1.64·37-s + 0.937·41-s + 1.21·43-s + 0.150·44-s + 0.884·46-s + 0.875·47-s − 3/7·49-s + 0.707·50-s + 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71478 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71478 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(71478\)    =    \(2 \cdot 3^{2} \cdot 11 \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{71478} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 71478,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.533856224$
$L(\frac12)$  $\approx$  $2.533856224$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−14.24267398377953, −13.80580112735200, −13.03693987036308, −12.48540647961038, −12.05034121024031, −11.46289104984701, −11.11458130868967, −10.62837896188704, −9.991567684597541, −9.563088879861302, −9.097240332391492, −8.321962526314512, −8.046393907817507, −7.655101541429240, −7.016533528224745, −6.264640214761086, −5.725668918300869, −5.528701350776218, −4.403076420261323, −3.999595171905160, −3.385447411134405, −2.543418476151216, −1.894966336716691, −1.208325596299773, −0.6659912715042692, 0.6659912715042692, 1.208325596299773, 1.894966336716691, 2.543418476151216, 3.385447411134405, 3.999595171905160, 4.403076420261323, 5.528701350776218, 5.725668918300869, 6.264640214761086, 7.016533528224745, 7.655101541429240, 8.046393907817507, 8.321962526314512, 9.097240332391492, 9.563088879861302, 9.991567684597541, 10.62837896188704, 11.11458130868967, 11.46289104984701, 12.05034121024031, 12.48540647961038, 13.03693987036308, 13.80580112735200, 14.24267398377953

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