Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 11 \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 + 2·5-s + 6-s − 7-s + 3·8-s + 9-s − 2·10-s + 11-s + 12-s + 6·13-s + 14-s − 2·15-s − 16-s − 2·17-s − 18-s − 4·19-s − 2·20-s + 21-s − 22-s − 3·24-s − 25-s − 6·26-s − 27-s + 28-s − 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.288·12-s + 1.66·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.218·21-s − 0.213·22-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122199 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122199 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(122199\)    =    \(3 \cdot 7 \cdot 11 \cdot 23^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{122199} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 122199,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
23 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−13.82219625728816, −13.18203879958063, −12.87092658806111, −12.62450928616991, −11.54483926151592, −11.36025295527716, −10.73250753352786, −10.35286424236161, −9.856806123772935, −9.450592278925831, −8.962429789466412, −8.393675128950850, −8.191414551603139, −7.331763703031171, −6.636924482058393, −6.403786281617044, −5.788784041130661, −5.397411396710102, −4.610335249182316, −4.086552548687128, −3.707407460293909, −2.750511490649688, −1.984271032672823, −1.400809249393244, −0.8405053847280338, 0, 0.8405053847280338, 1.400809249393244, 1.984271032672823, 2.750511490649688, 3.707407460293909, 4.086552548687128, 4.610335249182316, 5.397411396710102, 5.788784041130661, 6.403786281617044, 6.636924482058393, 7.331763703031171, 8.191414551603139, 8.393675128950850, 8.962429789466412, 9.450592278925831, 9.856806123772935, 10.35286424236161, 10.73250753352786, 11.36025295527716, 11.54483926151592, 12.62450928616991, 12.87092658806111, 13.18203879958063, 13.82219625728816

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