Properties

Degree 2
Conductor $ 2^{6} \cdot 5 \cdot 7^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5-s + 9-s + 2·13-s − 2·15-s + 6·17-s + 4·19-s − 6·23-s + 25-s − 4·27-s − 6·29-s − 4·31-s − 2·37-s + 4·39-s − 6·41-s − 10·43-s − 45-s − 6·47-s + 12·51-s + 6·53-s + 8·57-s − 12·59-s + 2·61-s − 2·65-s + 2·67-s − 12·69-s + 12·71-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.917·19-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s − 0.328·37-s + 0.640·39-s − 0.937·41-s − 1.52·43-s − 0.149·45-s − 0.875·47-s + 1.68·51-s + 0.824·53-s + 1.05·57-s − 1.56·59-s + 0.256·61-s − 0.248·65-s + 0.244·67-s − 1.44·69-s + 1.42·71-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15680\)    =    \(2^{6} \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{15680} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 15680,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
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

−16.16324922324037, −15.66860097395404, −15.02974040338461, −14.60694526781224, −14.09191007866598, −13.62550066443042, −13.08851812108073, −12.36511769128103, −11.76864738035183, −11.37450392637515, −10.51438366324048, −9.832649165314703, −9.508770456914171, −8.669911397736048, −8.274191454688856, −7.677245248941145, −7.311576544496534, −6.363921019068436, −5.604520816564666, −5.070090552922252, −3.952714963028514, −3.490471152450045, −3.108767963272630, −2.029796851088263, −1.366900260553527, 0, 1.366900260553527, 2.029796851088263, 3.108767963272630, 3.490471152450045, 3.952714963028514, 5.070090552922252, 5.604520816564666, 6.363921019068436, 7.311576544496534, 7.677245248941145, 8.274191454688856, 8.669911397736048, 9.508770456914171, 9.832649165314703, 10.51438366324048, 11.37450392637515, 11.76864738035183, 12.36511769128103, 13.08851812108073, 13.62550066443042, 14.09191007866598, 14.60694526781224, 15.02974040338461, 15.66860097395404, 16.16324922324037

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