Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 7-s + 9-s + 13-s − 15-s + 6·17-s + 4·19-s + 21-s + 25-s − 27-s + 6·29-s + 4·31-s − 35-s − 10·37-s − 39-s + 6·41-s − 8·43-s + 45-s + 49-s − 6·51-s − 6·53-s − 4·57-s + 12·59-s + 14·61-s − 63-s + 65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 1.45·17-s + 0.917·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.169·35-s − 1.64·37-s − 0.160·39-s + 0.937·41-s − 1.21·43-s + 0.149·45-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s + 1.79·61-s − 0.125·63-s + 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21840\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{21840} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 21840,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.262683179\)
\(L(\frac12)\)  \(\approx\)  \(2.262683179\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 - T \)
good11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.85525595489052, −15.01746286158449, −14.24488703510196, −14.07739282173831, −13.29688670177583, −12.85471834228741, −12.13813235288888, −11.85548315856070, −11.23854369765873, −10.38526270808427, −10.10453561056903, −9.659510020952594, −8.902860614465585, −8.238551338499772, −7.660962529103287, −6.843426313474370, −6.522221484293860, −5.679215723104720, −5.318190552029366, −4.690563782363292, −3.676073485541459, −3.230231720324850, −2.331796565072971, −1.329345860458418, −0.7032091526906143, 0.7032091526906143, 1.329345860458418, 2.331796565072971, 3.230231720324850, 3.676073485541459, 4.690563782363292, 5.318190552029366, 5.679215723104720, 6.522221484293860, 6.843426313474370, 7.660962529103287, 8.238551338499772, 8.902860614465585, 9.659510020952594, 10.10453561056903, 10.38526270808427, 11.23854369765873, 11.85548315856070, 12.13813235288888, 12.85471834228741, 13.29688670177583, 14.07739282173831, 14.24488703510196, 15.01746286158449, 15.85525595489052

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