Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 23 \cdot 29 $
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 + 9-s − 2·11-s + 2·13-s + 15-s − 4·17-s + 23-s + 25-s − 27-s + 29-s + 2·31-s + 2·33-s + 2·37-s − 2·39-s + 6·41-s + 4·43-s − 45-s − 7·49-s + 4·51-s − 10·53-s + 2·55-s + 10·59-s − 2·61-s − 2·65-s − 69-s − 10·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.258·15-s − 0.970·17-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s + 0.359·31-s + 0.348·33-s + 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 49-s + 0.560·51-s − 1.37·53-s + 0.269·55-s + 1.30·59-s − 0.256·61-s − 0.248·65-s − 0.120·69-s − 1.18·71-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 160080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 160080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(160080\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{160080} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 160080,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.085982640$
$L(\frac12)$  $\approx$  $1.085982640$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;23,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.24275746408438, −12.82783162433102, −12.32961464357928, −11.77448076517176, −11.35671794144766, −10.89651661548649, −10.62943706274777, −10.01518446255552, −9.425287397154051, −8.989068539575669, −8.390311064602453, −7.938559538309436, −7.465718456473718, −6.888577727672710, −6.352923839260818, −6.002392411299102, −5.303661091613818, −4.817688042295810, −4.306834106408254, −3.844491971030345, −3.061764544310201, −2.578056861533233, −1.808844267687734, −1.078047425602228, −0.3523202827421212, 0.3523202827421212, 1.078047425602228, 1.808844267687734, 2.578056861533233, 3.061764544310201, 3.844491971030345, 4.306834106408254, 4.817688042295810, 5.303661091613818, 6.002392411299102, 6.352923839260818, 6.888577727672710, 7.465718456473718, 7.938559538309436, 8.390311064602453, 8.989068539575669, 9.425287397154051, 10.01518446255552, 10.62943706274777, 10.89651661548649, 11.35671794144766, 11.77448076517176, 12.32961464357928, 12.82783162433102, 13.24275746408438

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