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 − 2·13-s + 14-s + 16-s − 6·17-s − 8·19-s − 20-s + 25-s + 2·26-s − 28-s + 6·29-s − 4·31-s − 32-s + 6·34-s + 35-s − 10·37-s + 8·38-s + 40-s − 6·41-s + 4·43-s + 49-s − 50-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 − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 1.83·19-s − 0.223·20-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.169·35-s − 1.64·37-s + 1.29·38-s + 0.158·40-s − 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.141·50-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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.47560546052195, −13.75417101141008, −13.16040965725709, −12.80552954537808, −12.20798891913802, −11.79601823582006, −11.22870838263641, −10.61939254070555, −10.38051902103015, −9.847503163317964, −9.025772558644869, −8.700803464539577, −8.442908751723554, −7.674957045143887, −7.029262414926956, −6.699292141367780, −6.306044039713448, −5.430406350311710, −4.858195745607065, −4.146238075988775, −3.740166700728578, −2.836794321319396, −2.264473573498121, −1.774657652852691, −0.6150694507258641, 0, 0.6150694507258641, 1.774657652852691, 2.264473573498121, 2.836794321319396, 3.740166700728578, 4.146238075988775, 4.858195745607065, 5.430406350311710, 6.306044039713448, 6.699292141367780, 7.029262414926956, 7.674957045143887, 8.442908751723554, 8.700803464539577, 9.025772558644869, 9.847503163317964, 10.38051902103015, 10.61939254070555, 11.22870838263641, 11.79601823582006, 12.20798891913802, 12.80552954537808, 13.16040965725709, 13.75417101141008, 14.47560546052195

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