Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 11 \cdot 41^{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

\[\begin{aligned}\Lambda(s)=\mathstrut & 388311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388311 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(388311\)    =    \(3 \cdot 7 \cdot 11 \cdot 41^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{388311} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 388311,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11,\;41\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
41 \( 1 \)
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 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
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

−12.68398236616487, −12.27250205259612, −11.80596892298137, −11.29450253390690, −10.78311308735331, −10.22436714065140, −9.849802135242569, −9.563345626595408, −9.068888741342917, −8.446672558781848, −8.322478530399413, −7.708999097706801, −7.421910783700563, −6.859848810813961, −6.471556086452847, −5.740505656731792, −5.012349949785583, −4.546668893823390, −4.222453516374849, −3.852285302939615, −3.016187935323812, −2.598924358639048, −2.031629779208123, −1.272615443053238, −0.5294934370820043, 0, 0.5294934370820043, 1.272615443053238, 2.031629779208123, 2.598924358639048, 3.016187935323812, 3.852285302939615, 4.222453516374849, 4.546668893823390, 5.012349949785583, 5.740505656731792, 6.471556086452847, 6.859848810813961, 7.421910783700563, 7.708999097706801, 8.322478530399413, 8.446672558781848, 9.068888741342917, 9.563345626595408, 9.849802135242569, 10.22436714065140, 10.78311308735331, 11.29450253390690, 11.80596892298137, 12.27250205259612, 12.68398236616487

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