Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 17 $
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 + 7-s + 8-s + 10-s − 6·11-s − 13-s + 14-s + 16-s − 17-s + 19-s + 20-s − 6·22-s + 4·23-s + 25-s − 26-s + 28-s + 8·29-s − 3·31-s + 32-s − 34-s + 35-s − 6·37-s + 38-s + 40-s + 3·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.80·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.229·19-s + 0.223·20-s − 1.27·22-s + 0.834·23-s + 1/5·25-s − 0.196·26-s + 0.188·28-s + 1.48·29-s − 0.538·31-s + 0.176·32-s − 0.171·34-s + 0.169·35-s − 0.986·37-s + 0.162·38-s + 0.158·40-s + 0.468·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(139230\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{139230} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 139230,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
good11 \( 1 + 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.63591390386107, −13.23713426321660, −12.72940190973678, −12.36346642610775, −11.86431559330682, −11.22277993809634, −10.63925240062930, −10.57104561120124, −9.922778192152255, −9.399350458865093, −8.541137902596185, −8.454261621062488, −7.552669726964427, −7.329072705639309, −6.772899037646347, −6.021218817679484, −5.618227303457792, −5.111185203895113, −4.703948703252687, −4.253101152028589, −3.200853990142891, −2.962712119766315, −2.340273844668392, −1.766728604018872, −0.9456203676272823, 0, 0.9456203676272823, 1.766728604018872, 2.340273844668392, 2.962712119766315, 3.200853990142891, 4.253101152028589, 4.703948703252687, 5.111185203895113, 5.618227303457792, 6.021218817679484, 6.772899037646347, 7.329072705639309, 7.552669726964427, 8.454261621062488, 8.541137902596185, 9.399350458865093, 9.922778192152255, 10.57104561120124, 10.63925240062930, 11.22277993809634, 11.86431559330682, 12.36346642610775, 12.72940190973678, 13.23713426321660, 13.63591390386107

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