Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 7^{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 + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 5·11-s + 12-s − 15-s + 16-s + 4·17-s + 18-s − 8·19-s − 20-s + 5·22-s − 4·23-s + 24-s − 4·25-s + 27-s − 5·29-s − 30-s − 3·31-s + 32-s + 5·33-s + 4·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s + 1.06·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s − 0.928·29-s − 0.182·30-s − 0.538·31-s + 0.176·32-s + 0.870·33-s + 0.685·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{294} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 294,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.212064895$
$L(\frac12)$  $\approx$  $2.212064895$
$L(\frac{3}{2})$   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 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−19.23214280843830, −19.09370740067605, −17.55657392027253, −16.77434081037205, −15.94382361087411, −14.89518661667148, −14.54773315496784, −13.66609837008990, −12.62108491431094, −11.98109925847864, −11.03780090339218, −9.907574124581067, −8.855984729142016, −7.871716855401770, −6.826695236016565, −5.814131140345095, −4.216435326604729, −3.630553849723318, −1.927955524734150, 1.927955524734150, 3.630553849723318, 4.216435326604729, 5.814131140345095, 6.826695236016565, 7.871716855401770, 8.855984729142016, 9.907574124581067, 11.03780090339218, 11.98109925847864, 12.62108491431094, 13.66609837008990, 14.54773315496784, 14.89518661667148, 15.94382361087411, 16.77434081037205, 17.55657392027253, 19.09370740067605, 19.23214280843830

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