Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 2·7-s + 9-s − 2·11-s + 2·13-s − 15-s + 5·17-s + 19-s − 2·21-s − 3·23-s + 25-s + 27-s − 9·29-s − 4·31-s − 2·33-s + 2·35-s − 11·37-s + 2·39-s + 6·41-s + 10·43-s − 45-s − 5·47-s − 3·49-s + 5·51-s − 12·53-s + 2·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.258·15-s + 1.21·17-s + 0.229·19-s − 0.436·21-s − 0.625·23-s + 1/5·25-s + 0.192·27-s − 1.67·29-s − 0.718·31-s − 0.348·33-s + 0.338·35-s − 1.80·37-s + 0.320·39-s + 0.937·41-s + 1.52·43-s − 0.149·45-s − 0.729·47-s − 3/7·49-s + 0.700·51-s − 1.64·53-s + 0.269·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;67\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
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

−7.927049305146293088566365336236, −7.56732797060158044294168761677, −6.72792694392625133977373261129, −5.79829426570502952920053645078, −5.17942099587592739275761992695, −3.88343150108370613657083791183, −3.53650869804877961254753431371, −2.62788869945592423404761525160, −1.47455333861761761457420263958, 0, 1.47455333861761761457420263958, 2.62788869945592423404761525160, 3.53650869804877961254753431371, 3.88343150108370613657083791183, 5.17942099587592739275761992695, 5.79829426570502952920053645078, 6.72792694392625133977373261129, 7.56732797060158044294168761677, 7.927049305146293088566365336236

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