Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 113 $
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 − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 2·11-s + 12-s + 7·13-s + 14-s − 15-s + 16-s − 3·17-s + 18-s + 6·19-s − 20-s + 21-s − 2·22-s + 3·23-s + 24-s − 4·25-s + 7·26-s + 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s + 1.94·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s − 0.426·22-s + 0.625·23-s + 0.204·24-s − 4/5·25-s + 1.37·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(678\)    =    \(2 \cdot 3 \cdot 113\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{678} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 678,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.794091462$
$L(\frac12)$  $\approx$  $2.794091462$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;113\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;113\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
113 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.84292610685950, −19.00498055296827, −18.23010126152448, −17.64875042986248, −16.22874435697085, −15.87497902462689, −15.27702629647128, −14.38117516428283, −13.52831442371621, −13.27705369270798, −12.17189760070046, −11.26013482365468, −10.82338865975589, −9.595599142728426, −8.566190107358154, −7.916224310992049, −6.981714672723312, −5.918590750180955, −4.896634322892550, −3.825981634947609, −3.052964317238349, −1.541760641800157, 1.541760641800157, 3.052964317238349, 3.825981634947609, 4.896634322892550, 5.918590750180955, 6.981714672723312, 7.916224310992049, 8.566190107358154, 9.595599142728426, 10.82338865975589, 11.26013482365468, 12.17189760070046, 13.27705369270798, 13.52831442371621, 14.38117516428283, 15.27702629647128, 15.87497902462689, 16.22874435697085, 17.64875042986248, 18.23010126152448, 19.00498055296827, 19.84292610685950

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