Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 23^{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  − 3-s − 2·4-s + 5-s − 7-s + 9-s − 3·11-s + 2·12-s − 6·13-s − 15-s + 4·16-s − 4·17-s − 19-s − 2·20-s + 21-s + 25-s − 27-s + 2·28-s − 2·29-s + 5·31-s + 3·33-s − 35-s − 2·36-s − 8·37-s + 6·39-s − 3·41-s + 2·43-s + 6·44-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.577·12-s − 1.66·13-s − 0.258·15-s + 16-s − 0.970·17-s − 0.229·19-s − 0.447·20-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 0.371·29-s + 0.898·31-s + 0.522·33-s − 0.169·35-s − 1/3·36-s − 1.31·37-s + 0.960·39-s − 0.468·41-s + 0.304·43-s + 0.904·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(55545\)    =    \(3 \cdot 5 \cdot 7 \cdot 23^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{55545} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 55545,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.68188763323955, −14.04528123236263, −13.52520676232017, −13.12211874517771, −12.72737621528571, −12.15803365692071, −11.79848392792713, −10.89556180391510, −10.45660560265741, −10.00555184848447, −9.605546941433328, −9.082791679232907, −8.420897090414840, −7.971276280442171, −7.080804328818161, −6.922895739304091, −6.024386530700859, −5.432129931350409, −5.082490947766462, −4.525884344687333, −3.995102365856942, −3.058584577431433, −2.468431064456761, −1.751929309945939, −0.5994833316985192, 0, 0.5994833316985192, 1.751929309945939, 2.468431064456761, 3.058584577431433, 3.995102365856942, 4.525884344687333, 5.082490947766462, 5.432129931350409, 6.024386530700859, 6.922895739304091, 7.080804328818161, 7.971276280442171, 8.420897090414840, 9.082791679232907, 9.605546941433328, 10.00555184848447, 10.45660560265741, 10.89556180391510, 11.79848392792713, 12.15803365692071, 12.72737621528571, 13.12211874517771, 13.52520676232017, 14.04528123236263, 14.68188763323955

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