Properties

Degree 2
Conductor $ 3 \cdot 19 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 4-s − 2·5-s + 6-s − 3·8-s + 9-s − 2·10-s − 12-s + 6·13-s − 2·15-s − 16-s − 6·17-s + 18-s − 19-s + 2·20-s + 4·23-s − 3·24-s − 25-s + 6·26-s + 27-s + 2·29-s − 2·30-s + 8·31-s + 5·32-s − 6·34-s − 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 1.66·13-s − 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.229·19-s + 0.447·20-s + 0.834·23-s − 0.612·24-s − 1/5·25-s + 1.17·26-s + 0.192·27-s + 0.371·29-s − 0.365·30-s + 1.43·31-s + 0.883·32-s − 1.02·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{57} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 57,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.085302272$
$L(\frac12)$  $\approx$  $1.085302272$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;19\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−19.37894635129668, −18.55444003973377, −17.46523974298726, −15.67579180944111, −15.35090650815042, −13.89424674132429, −13.30135307533368, −12.07987440084704, −10.84155613367777, −9.032116632714081, −8.263513432862467, −6.467875895673487, −4.569913210805974, −3.443655183627892, 3.443655183627892, 4.569913210805974, 6.467875895673487, 8.263513432862467, 9.032116632714081, 10.84155613367777, 12.07987440084704, 13.30135307533368, 13.89424674132429, 15.35090650815042, 15.67579180944111, 17.46523974298726, 18.55444003973377, 19.37894635129668

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