Properties

Degree 2
Conductor $ 2 \cdot 3 \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 − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 3·11-s − 12-s + 13-s − 14-s + 16-s + 8·17-s + 18-s + 7·19-s + 21-s − 3·22-s + 5·23-s − 24-s − 5·25-s + 26-s − 27-s − 28-s + 4·29-s + 9·31-s + 32-s + 3·33-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.94·17-s + 0.235·18-s + 1.60·19-s + 0.218·21-s − 0.639·22-s + 1.04·23-s − 0.204·24-s − 25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.742·29-s + 1.61·31-s + 0.176·32-s + 0.522·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(22386\)    =    \(2 \cdot 3 \cdot 7 \cdot 13 \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{22386} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 22386,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.353729214$
$L(\frac12)$  $\approx$  $3.353729214$
$L(\frac{3}{2})$   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 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 - T \)
41 \( 1 - T \)
good5 \( 1 + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−15.63088525191993, −15.03228482789210, −14.32049432476221, −13.81160821798243, −13.36302561979157, −12.84590268539604, −12.13040363756687, −11.82870869708056, −11.40965196868548, −10.41452711065932, −10.19475885248610, −9.722275593091395, −8.827363013598007, −8.021197323490986, −7.476062962184801, −7.105756114747912, −6.038027956852107, −5.896792392799467, −5.071059422412314, −4.786373479153028, −3.703649965731662, −3.173742039038568, −2.632344694858133, −1.387573640059049, −0.7488403991667836, 0.7488403991667836, 1.387573640059049, 2.632344694858133, 3.173742039038568, 3.703649965731662, 4.786373479153028, 5.071059422412314, 5.896792392799467, 6.038027956852107, 7.105756114747912, 7.476062962184801, 8.021197323490986, 8.827363013598007, 9.722275593091395, 10.19475885248610, 10.41452711065932, 11.40965196868548, 11.82870869708056, 12.13040363756687, 12.84590268539604, 13.36302561979157, 13.81160821798243, 14.32049432476221, 15.03228482789210, 15.63088525191993

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