Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 11-s − 2·13-s + 4·17-s + 6·19-s + 27-s − 8·29-s + 8·31-s + 33-s − 10·37-s − 2·39-s − 8·41-s + 2·43-s − 8·47-s + 4·51-s + 2·53-s + 6·57-s − 12·59-s − 10·61-s − 12·67-s + 8·71-s + 6·73-s − 2·79-s + 81-s + 16·83-s − 8·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.970·17-s + 1.37·19-s + 0.192·27-s − 1.48·29-s + 1.43·31-s + 0.174·33-s − 1.64·37-s − 0.320·39-s − 1.24·41-s + 0.304·43-s − 1.16·47-s + 0.560·51-s + 0.274·53-s + 0.794·57-s − 1.56·59-s − 1.28·61-s − 1.46·67-s + 0.949·71-s + 0.702·73-s − 0.225·79-s + 1/9·81-s + 1.75·83-s − 0.857·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(161700\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{161700} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 161700,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.59294768495248, −13.21337740624104, −12.30790164140251, −12.21275684073938, −11.75170181545840, −11.19323861897438, −10.42873104802864, −10.21468914579591, −9.562675462823217, −9.271644782766144, −8.799509463545553, −8.041812199898881, −7.745169586681408, −7.341626120946110, −6.723576951821327, −6.216691605136880, −5.534493422025651, −5.025454228658211, −4.650986534665620, −3.790637903840973, −3.237505039429869, −3.102316108399565, −2.098470447705659, −1.617332824509437, −0.9503100134573424, 0, 0.9503100134573424, 1.617332824509437, 2.098470447705659, 3.102316108399565, 3.237505039429869, 3.790637903840973, 4.650986534665620, 5.025454228658211, 5.534493422025651, 6.216691605136880, 6.723576951821327, 7.341626120946110, 7.745169586681408, 8.041812199898881, 8.799509463545553, 9.271644782766144, 9.562675462823217, 10.21468914579591, 10.42873104802864, 11.19323861897438, 11.75170181545840, 12.21275684073938, 12.30790164140251, 13.21337740624104, 13.59294768495248

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