Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 23^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s − 7-s + 9-s + 2·10-s + 5·11-s + 2·12-s − 2·14-s + 15-s − 4·16-s − 6·17-s + 2·18-s − 19-s + 2·20-s − 21-s + 10·22-s + 25-s + 27-s − 2·28-s + 2·30-s + 2·31-s − 8·32-s + 5·33-s − 12·34-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.632·10-s + 1.50·11-s + 0.577·12-s − 0.534·14-s + 0.258·15-s − 16-s − 1.45·17-s + 0.471·18-s − 0.229·19-s + 0.447·20-s − 0.218·21-s + 2.13·22-s + 1/5·25-s + 0.192·27-s − 0.377·28-s + 0.365·30-s + 0.359·31-s − 1.41·32-s + 0.870·33-s − 2.05·34-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  =  0
Selberg data  =  $(2,\ 55545,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.120607549$
$L(\frac12)$  $\approx$  $7.120607549$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
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

−14.29172209104356, −13.79660111294970, −13.58390421130560, −12.87541695653421, −12.70618431645889, −11.87139525400849, −11.64318288386118, −11.00254830908130, −10.32709863589034, −9.655494225910417, −9.265138466710789, −8.571879778227518, −8.467238402012267, −7.141229488962617, −6.810532976455211, −6.526300133985092, −5.800797445617041, −5.265952279566191, −4.532979059247825, −4.008264409096911, −3.701178044956409, −2.918828108144077, −2.279532799844134, −1.777368125640053, −0.6749607945580646, 0.6749607945580646, 1.777368125640053, 2.279532799844134, 2.918828108144077, 3.701178044956409, 4.008264409096911, 4.532979059247825, 5.265952279566191, 5.800797445617041, 6.526300133985092, 6.810532976455211, 7.141229488962617, 8.467238402012267, 8.571879778227518, 9.265138466710789, 9.655494225910417, 10.32709863589034, 11.00254830908130, 11.64318288386118, 11.87139525400849, 12.70618431645889, 12.87541695653421, 13.58390421130560, 13.79660111294970, 14.29172209104356

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