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-s − 3-s − 4-s + 5-s + 6-s + 7-s + 3·8-s + 9-s − 10-s − 4·11-s + 12-s − 6·13-s − 14-s − 15-s − 16-s + 2·17-s − 18-s − 20-s − 21-s + 4·22-s − 3·24-s + 25-s + 6·26-s − 27-s − 28-s + 6·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s − 1.66·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s − 0.218·21-s + 0.852·22-s − 0.612·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.182·30-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  =  $(2,\ 55545,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4723362981$
$L(\frac12)$  $\approx$  $0.4723362981$
$L(\frac{3}{2})$   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 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.41558624503797, −13.82105743641078, −13.46163520850108, −12.72394487925186, −12.40766567681559, −11.95437730509401, −11.10209235586590, −10.66955710074807, −10.19274840351801, −9.863851473013402, −9.344359883890006, −8.753782657472380, −8.052772297259610, −7.626644818608046, −7.319176557077920, −6.485626547185589, −5.798808377179880, −5.195694939631035, −4.725082742519246, −4.520174176206712, −3.346826162021896, −2.639072728370039, −1.950883194058826, −1.181451736868146, −0.3039228514515766, 0.3039228514515766, 1.181451736868146, 1.950883194058826, 2.639072728370039, 3.346826162021896, 4.520174176206712, 4.725082742519246, 5.195694939631035, 5.798808377179880, 6.485626547185589, 7.319176557077920, 7.626644818608046, 8.052772297259610, 8.753782657472380, 9.344359883890006, 9.863851473013402, 10.19274840351801, 10.66955710074807, 11.10209235586590, 11.95437730509401, 12.40766567681559, 12.72394487925186, 13.46163520850108, 13.82105743641078, 14.41558624503797

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