Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 7 \cdot 1259 $
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 + 7-s + 8-s + 9-s − 10-s − 2·11-s + 12-s + 14-s − 15-s + 16-s + 4·17-s + 18-s − 19-s − 20-s + 21-s − 2·22-s − 4·23-s + 24-s − 4·25-s + 27-s + 28-s + 2·29-s − 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 52878 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 52878 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(52878\)    =    \(2 \cdot 3 \cdot 7 \cdot 1259\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{52878} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 52878,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.057166572$
$L(\frac12)$  $\approx$  $4.057166572$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7,\;1259\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;1259\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 - T \)
1259 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
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

−14.39347889901706, −14.03450387015319, −13.48625811815046, −12.98237153416665, −12.46223605183439, −11.93443520990442, −11.62220455653280, −10.80355552731088, −10.42935244183438, −9.954720869329696, −9.155246194140816, −8.698778906241322, −7.987177090629848, −7.520396954975000, −7.364616544830440, −6.397344391394783, −5.782626866385418, −5.319751717952922, −4.666254727052464, −3.942605963663920, −3.636075788115923, −2.915705292411452, −2.169296774129008, −1.671519492314937, −0.5779751652877997, 0.5779751652877997, 1.671519492314937, 2.169296774129008, 2.915705292411452, 3.636075788115923, 3.942605963663920, 4.666254727052464, 5.319751717952922, 5.782626866385418, 6.397344391394783, 7.364616544830440, 7.520396954975000, 7.987177090629848, 8.698778906241322, 9.155246194140816, 9.954720869329696, 10.42935244183438, 10.80355552731088, 11.62220455653280, 11.93443520990442, 12.46223605183439, 12.98237153416665, 13.48625811815046, 14.03450387015319, 14.39347889901706

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