Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11^{2} $
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 + 4·13-s − 14-s + 16-s + 4·19-s + 20-s + 25-s + 4·26-s − 28-s − 6·29-s − 10·31-s + 32-s − 35-s + 2·37-s + 4·38-s + 40-s − 12·41-s + 4·43-s − 6·47-s + 49-s + 50-s + 4·52-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 + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.917·19-s + 0.223·20-s + 1/5·25-s + 0.784·26-s − 0.188·28-s − 1.11·29-s − 1.79·31-s + 0.176·32-s − 0.169·35-s + 0.328·37-s + 0.648·38-s + 0.158·40-s − 1.87·41-s + 0.609·43-s − 0.875·47-s + 1/7·49-s + 0.141·50-s + 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(76230\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{76230} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 76230,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
good13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.16703621310519, −13.76757709785361, −13.21826797062161, −13.00203903765169, −12.48012701818914, −11.70973654271050, −11.42330671246764, −10.91414207066586, −10.30221750808553, −9.858459173047993, −9.241461516720233, −8.781093971909224, −8.204707981406857, −7.409392429722529, −7.093283909920494, −6.471036927517497, −5.802055165835509, −5.567938396792377, −4.973597981739713, −4.175061359907067, −3.527469443559780, −3.313201708961279, −2.406429116881263, −1.732127893931715, −1.146771499001499, 0, 1.146771499001499, 1.732127893931715, 2.406429116881263, 3.313201708961279, 3.527469443559780, 4.175061359907067, 4.973597981739713, 5.567938396792377, 5.802055165835509, 6.471036927517497, 7.093283909920494, 7.409392429722529, 8.204707981406857, 8.781093971909224, 9.241461516720233, 9.858459173047993, 10.30221750808553, 10.91414207066586, 11.42330671246764, 11.70973654271050, 12.48012701818914, 13.00203903765169, 13.21826797062161, 13.76757709785361, 14.16703621310519

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