Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 13 \cdot 29 $
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-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 13-s + 14-s − 15-s + 16-s − 3·17-s + 18-s − 19-s − 20-s + 21-s + 22-s + 3·23-s + 24-s − 4·25-s − 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 + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.625·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(24882\)    =    \(2 \cdot 3 \cdot 11 \cdot 13 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{24882} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 24882,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;13,\;29\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;13,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−15.41774792116834, −15.00397338283526, −14.63337523832411, −14.11507898612722, −13.47761463615517, −12.94934844472169, −12.68140116941841, −11.70688893647750, −11.45939504478501, −11.03631433562606, −10.03625775621487, −9.811087453551154, −8.817287715094643, −8.536232454775946, −7.663852861567746, −7.437885720319620, −6.543255027867653, −6.183723339719197, −5.190579888437125, −4.667750370866134, −4.144880570804156, −3.446003408066807, −2.832755153440247, −2.028714624674828, −1.355629463398903, 0, 1.355629463398903, 2.028714624674828, 2.832755153440247, 3.446003408066807, 4.144880570804156, 4.667750370866134, 5.190579888437125, 6.183723339719197, 6.543255027867653, 7.437885720319620, 7.663852861567746, 8.536232454775946, 8.817287715094643, 9.811087453551154, 10.03625775621487, 11.03631433562606, 11.45939504478501, 11.70688893647750, 12.68140116941841, 12.94934844472169, 13.47761463615517, 14.11507898612722, 14.63337523832411, 15.00397338283526, 15.41774792116834

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