Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 7 \cdot 13 \cdot 41 $
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 + 4-s + 5-s + 7-s − 8-s − 10-s + 2·11-s − 13-s − 14-s + 16-s − 4·17-s − 8·19-s + 20-s − 2·22-s − 3·23-s − 4·25-s + 26-s + 28-s − 2·29-s + 4·31-s − 32-s + 4·34-s + 35-s + 3·37-s + 8·38-s − 40-s + 41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.603·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s − 1.83·19-s + 0.223·20-s − 0.426·22-s − 0.625·23-s − 4/5·25-s + 0.196·26-s + 0.188·28-s − 0.371·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s + 0.169·35-s + 0.493·37-s + 1.29·38-s − 0.158·40-s + 0.156·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(67158\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13 \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{67158} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 67158,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9711431584$
$L(\frac12)$  $\approx$  $0.9711431584$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;13,\;41\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;13,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 - T \)
13 \( 1 + T \)
41 \( 1 - T \)
good5 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
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

−14.21955945881023, −13.70533919639770, −13.17519798866281, −12.64007910060042, −12.14184050337584, −11.42303668466859, −11.23996787407396, −10.59803153075608, −9.974697998531603, −9.743423233993373, −8.965408307357097, −8.646446135122707, −8.092125235255504, −7.617881423116481, −6.769313676089706, −6.486653302273445, −5.997900147428935, −5.298375629512218, −4.452754450607084, −4.171973684460400, −3.320351367718180, −2.359376390293096, −2.059029809943333, −1.419778385497533, −0.3579845067611262, 0.3579845067611262, 1.419778385497533, 2.059029809943333, 2.359376390293096, 3.320351367718180, 4.171973684460400, 4.452754450607084, 5.298375629512218, 5.997900147428935, 6.486653302273445, 6.769313676089706, 7.617881423116481, 8.092125235255504, 8.646446135122707, 8.965408307357097, 9.743423233993373, 9.974697998531603, 10.59803153075608, 11.23996787407396, 11.42303668466859, 12.14184050337584, 12.64007910060042, 13.17519798866281, 13.70533919639770, 14.21955945881023

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