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  + 3-s − 2·4-s − 5-s + 7-s + 9-s − 4·11-s − 2·12-s + 3·13-s − 15-s + 4·16-s + 17-s + 2·20-s + 21-s + 25-s + 27-s − 2·28-s − 3·29-s − 6·31-s − 4·33-s − 35-s − 2·36-s + 6·37-s + 3·39-s + 12·41-s − 6·43-s + 8·44-s − 45-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.577·12-s + 0.832·13-s − 0.258·15-s + 16-s + 0.242·17-s + 0.447·20-s + 0.218·21-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.557·29-s − 1.07·31-s − 0.696·33-s − 0.169·35-s − 1/3·36-s + 0.986·37-s + 0.480·39-s + 1.87·41-s − 0.914·43-s + 1.20·44-s − 0.149·45-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  =  $(2,\ 55545,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.690841743$
$L(\frac12)$  $\approx$  $1.690841743$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−14.41002510537340, −13.73953394194163, −13.54697206442297, −12.86850684037288, −12.59932624469724, −12.00726510891917, −11.15447122506921, −10.74203920916458, −10.43091839156666, −9.418307359817429, −9.327523701848946, −8.706656061695909, −7.972188077746858, −7.836096751793204, −7.395397937646047, −6.386787293953621, −5.737274866663499, −5.265149991810291, −4.570783819620774, −4.123716380207762, −3.489887429282206, −2.946574395718199, −2.137687326361583, −1.271862350640893, −0.4717942214636832, 0.4717942214636832, 1.271862350640893, 2.137687326361583, 2.946574395718199, 3.489887429282206, 4.123716380207762, 4.570783819620774, 5.265149991810291, 5.737274866663499, 6.386787293953621, 7.395397937646047, 7.836096751793204, 7.972188077746858, 8.706656061695909, 9.327523701848946, 9.418307359817429, 10.43091839156666, 10.74203920916458, 11.15447122506921, 12.00726510891917, 12.59932624469724, 12.86850684037288, 13.54697206442297, 13.73953394194163, 14.41002510537340

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