Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 73 \cdot 137 $
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 − 2·3-s + 4-s + 5-s − 2·6-s − 7-s + 8-s + 9-s + 10-s + 2·11-s − 2·12-s + 5·13-s − 14-s − 2·15-s + 16-s − 6·17-s + 18-s − 4·19-s + 20-s + 2·21-s + 2·22-s − 4·23-s − 2·24-s + 25-s + 5·26-s + 4·27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.577·12-s + 1.38·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.436·21-s + 0.426·22-s − 0.834·23-s − 0.408·24-s + 1/5·25-s + 0.980·26-s + 0.769·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100010\)    =    \(2 \cdot 5 \cdot 73 \cdot 137\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100010} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 100010,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;73,\;137\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;73,\;137\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
73 \( 1 - T \)
137 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.79642479575599, −13.38396152182145, −13.13923240714675, −12.56580669867426, −11.95031429479375, −11.47557508687197, −11.36041104303729, −10.50432865601724, −10.41458538536863, −9.753123892900664, −8.883128222367942, −8.634514974035561, −8.075380440235685, −7.062682119155321, −6.572716100619171, −6.334012138581204, −6.024647605339594, −5.339485009721814, −4.821338363389602, −4.134776346461294, −3.858324916631525, −2.989343432551428, −2.292607116546566, −1.641448114598130, −0.8785573999052248, 0, 0.8785573999052248, 1.641448114598130, 2.292607116546566, 2.989343432551428, 3.858324916631525, 4.134776346461294, 4.821338363389602, 5.339485009721814, 6.024647605339594, 6.334012138581204, 6.572716100619171, 7.062682119155321, 8.075380440235685, 8.634514974035561, 8.883128222367942, 9.753123892900664, 10.41458538536863, 10.50432865601724, 11.36041104303729, 11.47557508687197, 11.95031429479375, 12.56580669867426, 13.13923240714675, 13.38396152182145, 13.79642479575599

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