Properties

Degree 2
Conductor $ 2^{6} \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

\[\begin{aligned}\Lambda(s)=\mathstrut & 87360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(87360\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{87360} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 87360,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.035143623$
$L(\frac12)$  $\approx$  $2.035143623$
$L(\frac{3}{2})$   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

−13.88381859153427, −13.32509536617211, −12.89695512573916, −12.14931135108279, −12.07079831635317, −11.37115660742446, −11.10491388569920, −10.46629585290129, −9.938487081977243, −9.425771046265241, −9.050618960712025, −8.051950656097775, −7.863646917439189, −7.369463842575492, −6.842576709276161, −6.090592901910933, −5.549900379663710, −5.209990760336431, −4.578098959498429, −3.870699926453742, −3.446837005464409, −2.699849229531420, −1.894148237009959, −1.126566025715894, −0.5448283700703436, 0.5448283700703436, 1.126566025715894, 1.894148237009959, 2.699849229531420, 3.446837005464409, 3.870699926453742, 4.578098959498429, 5.209990760336431, 5.549900379663710, 6.090592901910933, 6.842576709276161, 7.369463842575492, 7.863646917439189, 8.051950656097775, 9.050618960712025, 9.425771046265241, 9.938487081977243, 10.46629585290129, 11.10491388569920, 11.37115660742446, 12.07079831635317, 12.14931135108279, 12.89695512573916, 13.32509536617211, 13.88381859153427

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