Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 73 \cdot 137 $
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 − 2·9-s + 10-s + 3·11-s + 12-s + 4·13-s + 14-s + 15-s + 16-s − 6·17-s − 2·18-s + 20-s + 21-s + 3·22-s + 6·23-s + 24-s + 25-s + 4·26-s − 5·27-s + 28-s + 3·29-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 − 2/3·9-s + 0.316·10-s + 0.904·11-s + 0.288·12-s + 1.10·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.471·18-s + 0.223·20-s + 0.218·21-s + 0.639·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.962·27-s + 0.188·28-s + 0.557·29-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  =  0
Selberg data  =  $(2,\ 100010,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.154922010$
$L(\frac12)$  $\approx$  $7.154922010$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
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

−13.73419874569855, −13.34443642737169, −13.07135398254662, −12.28373162107120, −11.87694398011219, −11.17988868391088, −10.91638604851543, −10.67985229516301, −9.568007469818122, −9.215089868787024, −8.825790029517254, −8.367285186909627, −7.767813739028613, −7.054876537673783, −6.533567425524535, −6.228473619442083, −5.467691724302790, −5.127608260796243, −4.258052611607661, −3.927598683907679, −3.311988096050527, −2.599989308009612, −2.180038751762173, −1.437856372192879, −0.7243495325707646, 0.7243495325707646, 1.437856372192879, 2.180038751762173, 2.599989308009612, 3.311988096050527, 3.927598683907679, 4.258052611607661, 5.127608260796243, 5.467691724302790, 6.228473619442083, 6.533567425524535, 7.054876537673783, 7.767813739028613, 8.367285186909627, 8.825790029517254, 9.215089868787024, 9.568007469818122, 10.67985229516301, 10.91638604851543, 11.17988868391088, 11.87694398011219, 12.28373162107120, 13.07135398254662, 13.34443642737169, 13.73419874569855

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