Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5 \cdot 67 $
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 − 0.212·7-s + 9-s − 2.52·11-s − 3.33·13-s − 15-s − 7.70·17-s + 7.01·19-s − 0.212·21-s − 1.74·23-s + 25-s + 27-s + 1.69·29-s − 2.26·31-s − 2.52·33-s + 0.212·35-s + 9.86·37-s − 3.33·39-s + 4.26·41-s − 1.58·43-s − 45-s − 5.44·47-s − 6.95·49-s − 7.70·51-s + 6.91·53-s + 2.52·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.0802·7-s + 0.333·9-s − 0.762·11-s − 0.923·13-s − 0.258·15-s − 1.86·17-s + 1.60·19-s − 0.0463·21-s − 0.363·23-s + 0.200·25-s + 0.192·27-s + 0.314·29-s − 0.407·31-s − 0.440·33-s + 0.0359·35-s + 1.62·37-s − 0.533·39-s + 0.666·41-s − 0.242·43-s − 0.149·45-s − 0.793·47-s − 0.993·49-s − 1.07·51-s + 0.949·53-s + 0.341·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8040} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8040,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.657625693$
$L(\frac12)$  $\approx$  $1.657625693$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;67\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;67\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
67 \( 1 - T \)
good7 \( 1 + 0.212T + 7T^{2} \)
11 \( 1 + 2.52T + 11T^{2} \)
13 \( 1 + 3.33T + 13T^{2} \)
17 \( 1 + 7.70T + 17T^{2} \)
19 \( 1 - 7.01T + 19T^{2} \)
23 \( 1 + 1.74T + 23T^{2} \)
29 \( 1 - 1.69T + 29T^{2} \)
31 \( 1 + 2.26T + 31T^{2} \)
37 \( 1 - 9.86T + 37T^{2} \)
41 \( 1 - 4.26T + 41T^{2} \)
43 \( 1 + 1.58T + 43T^{2} \)
47 \( 1 + 5.44T + 47T^{2} \)
53 \( 1 - 6.91T + 53T^{2} \)
59 \( 1 + 7.97T + 59T^{2} \)
61 \( 1 - 8.27T + 61T^{2} \)
71 \( 1 - 4.11T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + 6.71T + 79T^{2} \)
83 \( 1 - 6.07T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 - 7.28T + 97T^{2} \)
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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

−7.83571642177540022026815082303, −7.26156377893633765481780814573, −6.64931900407402984229321026781, −5.70244773187608154477416725364, −4.83591989818183792963083344930, −4.38134817709042252580172667534, −3.39056787515570647044632171492, −2.68072120223795021521353047163, −2.00053063930800597301056121365, −0.59250365428848048760976506317, 0.59250365428848048760976506317, 2.00053063930800597301056121365, 2.68072120223795021521353047163, 3.39056787515570647044632171492, 4.38134817709042252580172667534, 4.83591989818183792963083344930, 5.70244773187608154477416725364, 6.64931900407402984229321026781, 7.26156377893633765481780814573, 7.83571642177540022026815082303

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