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 1

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  =  1
Selberg data  =  $(2,\ 87360,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]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

−14.30462240833486, −13.66212526959516, −13.11372171502676, −12.63706943177701, −12.32480945909852, −11.74850249442593, −11.15085084004193, −10.62892795174243, −10.14732667110227, −9.574086539395872, −9.168017023022166, −8.688202961546197, −7.882653377615889, −7.679693620016952, −7.288494255930620, −6.386579359355896, −6.025607402545435, −5.426080795868326, −4.595423211002702, −4.151321400040806, −3.655678521609873, −2.819308997781779, −2.634134324313613, −1.615358735206437, −0.9381703478144893, 0, 0.9381703478144893, 1.615358735206437, 2.634134324313613, 2.819308997781779, 3.655678521609873, 4.151321400040806, 4.595423211002702, 5.426080795868326, 6.025607402545435, 6.386579359355896, 7.288494255930620, 7.679693620016952, 7.882653377615889, 8.688202961546197, 9.168017023022166, 9.574086539395872, 10.14732667110227, 10.62892795174243, 11.15085084004193, 11.74850249442593, 12.32480945909852, 12.63706943177701, 13.11372171502676, 13.66212526959516, 14.30462240833486

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