Properties

Degree 2
Conductor $ 2^{3} \cdot 5 \cdot 17^{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  − 5-s + 4·7-s − 3·9-s − 4·11-s − 2·13-s + 4·19-s − 4·23-s + 25-s + 2·29-s + 8·31-s − 4·35-s − 6·37-s + 6·41-s − 8·43-s + 3·45-s + 4·47-s + 9·49-s + 6·53-s + 4·55-s − 4·59-s + 2·61-s − 12·63-s + 2·65-s + 8·67-s + 6·73-s − 16·77-s + 9·81-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s − 9-s − 1.20·11-s − 0.554·13-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.937·41-s − 1.21·43-s + 0.447·45-s + 0.583·47-s + 9/7·49-s + 0.824·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s − 1.51·63-s + 0.248·65-s + 0.977·67-s + 0.702·73-s − 1.82·77-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11560\)    =    \(2^{3} \cdot 5 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{11560} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 11560,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
17 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.85565459613921, −15.94907974512770, −15.63013963060837, −15.00025323049627, −14.37489405493841, −13.98030803879819, −13.50606887357528, −12.56039397374837, −11.94004607954643, −11.61203636283116, −11.05224207911734, −10.37527924964385, −9.899454437662760, −8.872122818424991, −8.352940208848357, −7.878962096579196, −7.502377558738376, −6.569899671408341, −5.610857668712367, −5.159934176484283, −4.669953786442642, −3.761588325266893, −2.781697958503439, −2.287295016865806, −1.144388247215630, 0, 1.144388247215630, 2.287295016865806, 2.781697958503439, 3.761588325266893, 4.669953786442642, 5.159934176484283, 5.610857668712367, 6.569899671408341, 7.502377558738376, 7.878962096579196, 8.352940208848357, 8.872122818424991, 9.899454437662760, 10.37527924964385, 11.05224207911734, 11.61203636283116, 11.94004607954643, 12.56039397374837, 13.50606887357528, 13.98030803879819, 14.37489405493841, 15.00025323049627, 15.63013963060837, 15.94907974512770, 16.85565459613921

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