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 − 2.80·7-s + 9-s + 0.770·11-s − 3.59·13-s − 15-s − 0.618·17-s − 6.93·19-s − 2.80·21-s + 1.25·23-s + 25-s + 27-s + 3.30·29-s + 3.43·31-s + 0.770·33-s + 2.80·35-s − 5.13·37-s − 3.59·39-s − 1.43·41-s − 4.84·43-s − 45-s + 10.8·47-s + 0.840·49-s − 0.618·51-s + 12.6·53-s − 0.770·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.05·7-s + 0.333·9-s + 0.232·11-s − 0.995·13-s − 0.258·15-s − 0.149·17-s − 1.59·19-s − 0.611·21-s + 0.261·23-s + 0.200·25-s + 0.192·27-s + 0.613·29-s + 0.616·31-s + 0.134·33-s + 0.473·35-s − 0.843·37-s − 0.574·39-s − 0.223·41-s − 0.738·43-s − 0.149·45-s + 1.58·47-s + 0.120·49-s − 0.0865·51-s + 1.74·53-s − 0.103·55-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  =  $(2,\ 8040,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.362415592$
$L(\frac12)$  $\approx$  $1.362415592$
$L(\frac{3}{2})$   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 \( 1 + 2.80T + 7T^{2} \)
11 \( 1 - 0.770T + 11T^{2} \)
13 \( 1 + 3.59T + 13T^{2} \)
17 \( 1 + 0.618T + 17T^{2} \)
19 \( 1 + 6.93T + 19T^{2} \)
23 \( 1 - 1.25T + 23T^{2} \)
29 \( 1 - 3.30T + 29T^{2} \)
31 \( 1 - 3.43T + 31T^{2} \)
37 \( 1 + 5.13T + 37T^{2} \)
41 \( 1 + 1.43T + 41T^{2} \)
43 \( 1 + 4.84T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 - 0.397T + 59T^{2} \)
61 \( 1 + 12.9T + 61T^{2} \)
71 \( 1 + 1.10T + 71T^{2} \)
73 \( 1 + 2.18T + 73T^{2} \)
79 \( 1 - 9.90T + 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 - 18.2T + 97T^{2} \)
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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

−7.81120253275943434040387587568, −7.08106593339965020959308674593, −6.63630888328890681323398488943, −5.89334191035943005024976557210, −4.81658561376792760534465004097, −4.23740032555640320611370875275, −3.44435859095274915156808371700, −2.73900783598996625954076472781, −1.98144757096651863868258708814, −0.53318340533996517484602872159, 0.53318340533996517484602872159, 1.98144757096651863868258708814, 2.73900783598996625954076472781, 3.44435859095274915156808371700, 4.23740032555640320611370875275, 4.81658561376792760534465004097, 5.89334191035943005024976557210, 6.63630888328890681323398488943, 7.08106593339965020959308674593, 7.81120253275943434040387587568

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