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 + 4-s + 5-s + 8-s − 3·9-s + 10-s + 4·11-s + 6·13-s + 16-s − 2·17-s − 3·18-s + 4·19-s + 20-s + 4·22-s + 2·23-s + 25-s + 6·26-s − 10·29-s − 8·31-s + 32-s − 2·34-s − 3·36-s + 10·37-s + 4·38-s + 40-s − 6·41-s − 6·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s − 9-s + 0.316·10-s + 1.20·11-s + 1.66·13-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.917·19-s + 0.223·20-s + 0.852·22-s + 0.417·23-s + 1/5·25-s + 1.17·26-s − 1.85·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s + 1.64·37-s + 0.648·38-s + 0.158·40-s − 0.937·41-s − 0.914·43-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 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.90045711243297, −13.41896181590764, −13.22876911371604, −12.73817180731958, −11.88446301643248, −11.44323090182202, −11.29355075991724, −10.86309698527572, −10.06199595036346, −9.430920470282645, −9.076269111551880, −8.601523354550034, −8.042340484738109, −7.317782426454501, −6.801108541114982, −6.213877090811198, −5.884780688042316, −5.421945562567906, −4.795337927401201, −4.037717783328260, −3.435669292641819, −3.295616682213946, −2.321023292765027, −1.604370107970545, −1.191754201292775, 0, 1.191754201292775, 1.604370107970545, 2.321023292765027, 3.295616682213946, 3.435669292641819, 4.037717783328260, 4.795337927401201, 5.421945562567906, 5.884780688042316, 6.213877090811198, 6.801108541114982, 7.317782426454501, 8.042340484738109, 8.601523354550034, 9.076269111551880, 9.430920470282645, 10.06199595036346, 10.86309698527572, 11.29355075991724, 11.44323090182202, 11.88446301643248, 12.73817180731958, 13.22876911371604, 13.41896181590764, 13.90045711243297

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