Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 17 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 2·7-s + 9-s + 13-s + 15-s − 17-s + 2·19-s − 2·21-s + 8·23-s + 25-s − 27-s − 6·29-s + 4·31-s − 2·35-s − 2·37-s − 39-s + 6·41-s − 2·43-s − 45-s − 3·49-s + 51-s + 2·53-s − 2·57-s − 14·59-s + 6·61-s + 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.458·19-s − 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.338·35-s − 0.328·37-s − 0.160·39-s + 0.937·41-s − 0.304·43-s − 0.149·45-s − 3/7·49-s + 0.140·51-s + 0.274·53-s − 0.264·57-s − 1.82·59-s + 0.768·61-s + 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(212160\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{212160} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 212160,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.09181837629378, −12.81585890458504, −12.25315901204820, −11.70369946565707, −11.30007092007641, −11.09803223856219, −10.60212416497732, −10.01852096525235, −9.497970016318572, −8.876447637722840, −8.639461876023771, −7.877790955303740, −7.556585539562594, −7.041338119713221, −6.604200037028158, −5.918118006884320, −5.458784534686778, −4.988766293461211, −4.423091611753043, −4.112454708900475, −3.204310594474956, −2.925379715666325, −1.969197901647935, −1.392495682071304, −0.8347130220571640, 0, 0.8347130220571640, 1.392495682071304, 1.969197901647935, 2.925379715666325, 3.204310594474956, 4.112454708900475, 4.423091611753043, 4.988766293461211, 5.458784534686778, 5.918118006884320, 6.604200037028158, 7.041338119713221, 7.556585539562594, 7.877790955303740, 8.639461876023771, 8.876447637722840, 9.497970016318572, 10.01852096525235, 10.60212416497732, 11.09803223856219, 11.30007092007641, 11.70369946565707, 12.25315901204820, 12.81585890458504, 13.09181837629378

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