Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 4091 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 3·5-s + 6-s − 5·7-s − 8-s + 9-s + 3·10-s − 5·11-s − 12-s − 6·13-s + 5·14-s + 3·15-s + 16-s − 4·17-s − 18-s − 6·19-s − 3·20-s + 5·21-s + 5·22-s − 9·23-s + 24-s + 4·25-s + 6·26-s − 27-s − 5·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 1.50·11-s − 0.288·12-s − 1.66·13-s + 1.33·14-s + 0.774·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 1.37·19-s − 0.670·20-s + 1.09·21-s + 1.06·22-s − 1.87·23-s + 0.204·24-s + 4/5·25-s + 1.17·26-s − 0.192·27-s − 0.944·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(24546\)    =    \(2 \cdot 3 \cdot 4091\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{24546} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 24546,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;4091\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;4091\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
4091 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−16.31528206139947, −15.67280215818628, −15.27765806666043, −15.13764688693100, −14.04077813169368, −13.21607344797135, −12.70678039238815, −12.43186175997173, −11.99957553692154, −11.12802351472510, −10.79347345852382, −10.08832008075510, −9.824393236399667, −9.136281439573843, −8.404230195222007, −7.775440278968898, −7.313595578356126, −6.865557401328231, −6.181236874744949, −5.581468262683396, −4.737631748021957, −4.009415906944245, −3.461691784247171, −2.526933557444731, −2.093900260196079, 0, 0, 0, 2.093900260196079, 2.526933557444731, 3.461691784247171, 4.009415906944245, 4.737631748021957, 5.581468262683396, 6.181236874744949, 6.865557401328231, 7.313595578356126, 7.775440278968898, 8.404230195222007, 9.136281439573843, 9.824393236399667, 10.08832008075510, 10.79347345852382, 11.12802351472510, 11.99957553692154, 12.43186175997173, 12.70678039238815, 13.21607344797135, 14.04077813169368, 15.13764688693100, 15.27765806666043, 15.67280215818628, 16.31528206139947

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