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  + 2-s + 3-s − 4-s − 5-s + 6-s − 7-s − 3·8-s + 9-s − 10-s − 12-s − 6·13-s − 14-s − 15-s − 16-s − 2·17-s + 18-s + 8·19-s + 20-s − 21-s − 3·24-s + 25-s − 6·26-s + 27-s + 28-s − 2·29-s − 30-s + 4·31-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 − 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 + 1.83·19-s + 0.223·20-s − 0.218·21-s − 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.182·30-s + 0.718·31-s + ⋯

Functional equation

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

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 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.58003990179407, −14.14643208291135, −13.64513682053911, −13.22712775412256, −12.71811308425447, −12.14032151208372, −11.82798048467272, −11.36888783394944, −10.29358187451852, −10.02642414126790, −9.348370047330654, −9.161511482376572, −8.390011093702788, −7.840342444117310, −7.270990171136421, −6.892431849307683, −6.030303025856026, −5.454600388976996, −4.767198814607318, −4.552817773607729, −3.733653270756340, −3.123646354162737, −2.824149464454914, −1.949071054053506, −0.8371493983340824, 0, 0.8371493983340824, 1.949071054053506, 2.824149464454914, 3.123646354162737, 3.733653270756340, 4.552817773607729, 4.767198814607318, 5.454600388976996, 6.030303025856026, 6.892431849307683, 7.270990171136421, 7.840342444117310, 8.390011093702788, 9.161511482376572, 9.348370047330654, 10.02642414126790, 10.29358187451852, 11.36888783394944, 11.82798048467272, 12.14032151208372, 12.71811308425447, 13.22712775412256, 13.64513682053911, 14.14643208291135, 14.58003990179407

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