Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s − 3·5-s + 2·6-s − 5·7-s + 9-s + 6·10-s + 11-s − 2·12-s + 2·13-s + 10·14-s + 3·15-s − 4·16-s − 17-s − 2·18-s − 19-s − 6·20-s + 5·21-s − 2·22-s − 4·23-s + 4·25-s − 4·26-s − 27-s − 10·28-s − 2·29-s − 6·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 1.34·5-s + 0.816·6-s − 1.88·7-s + 1/3·9-s + 1.89·10-s + 0.301·11-s − 0.577·12-s + 0.554·13-s + 2.67·14-s + 0.774·15-s − 16-s − 0.242·17-s − 0.471·18-s − 0.229·19-s − 1.34·20-s + 1.09·21-s − 0.426·22-s − 0.834·23-s + 4/5·25-s − 0.784·26-s − 0.192·27-s − 1.88·28-s − 0.371·29-s − 1.09·30-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  :  $\chi_{57} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 57,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;19\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.37915837374799681769563575875, −13.25936914052168015343287102801, −12.10480291153961981228299085685, −11.00179000913096952209865162760, −9.914070570189995801171942379165, −8.836427381447004052087018681826, −7.47848778675296589128891353641, −6.41173993136661837405396202601, −3.81591641511052084615077615666, 0, 3.81591641511052084615077615666, 6.41173993136661837405396202601, 7.47848778675296589128891353641, 8.836427381447004052087018681826, 9.914070570189995801171942379165, 11.00179000913096952209865162760, 12.10480291153961981228299085685, 13.25936914052168015343287102801, 15.37915837374799681769563575875

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