Properties

Degree 2
Conductor $ 3 \cdot 11 \cdot 13^{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 − 4·7-s + 3·8-s + 9-s − 2·10-s − 11-s + 12-s + 4·14-s − 2·15-s − 16-s − 2·17-s − 18-s − 2·20-s + 4·21-s + 22-s + 8·23-s − 3·24-s − 25-s − 27-s + 4·28-s − 6·29-s + 2·30-s + 8·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s + 1.06·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.447·20-s + 0.872·21-s + 0.213·22-s + 1.66·23-s − 0.612·24-s − 1/5·25-s − 0.192·27-s + 0.755·28-s − 1.11·29-s + 0.365·30-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5577\)    =    \(3 \cdot 11 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{5577} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 5577,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−17.78358871527210, −17.18385709311935, −16.95747227476380, −16.23869971097477, −15.69119140570058, −14.99874369760047, −14.02297111818638, −13.45501739756308, −12.99359328679928, −12.70138392501538, −11.62410559173780, −10.89922244330820, −10.17294629402426, −9.873900286897792, −9.251643371505200, −8.768985116389245, −7.828312590303411, −6.876255668074137, −6.571121455120252, −5.590926681676826, −5.108301475173027, −4.114324054020075, −3.209673627047249, −2.184907836093431, −1.011909030517971, 0, 1.011909030517971, 2.184907836093431, 3.209673627047249, 4.114324054020075, 5.108301475173027, 5.590926681676826, 6.571121455120252, 6.876255668074137, 7.828312590303411, 8.768985116389245, 9.251643371505200, 9.873900286897792, 10.17294629402426, 10.89922244330820, 11.62410559173780, 12.70138392501538, 12.99359328679928, 13.45501739756308, 14.02297111818638, 14.99874369760047, 15.69119140570058, 16.23869971097477, 16.95747227476380, 17.18385709311935, 17.78358871527210

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