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 + 4.52·7-s + 9-s + 3.01·11-s − 4.99·13-s − 15-s + 4.07·17-s + 1.78·19-s + 4.52·21-s + 4.12·23-s + 25-s + 27-s + 2.71·29-s + 4.66·31-s + 3.01·33-s − 4.52·35-s + 4.96·37-s − 4.99·39-s − 2.66·41-s − 9.11·43-s − 45-s − 2.72·47-s + 13.5·49-s + 4.07·51-s + 0.311·53-s − 3.01·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.71·7-s + 0.333·9-s + 0.909·11-s − 1.38·13-s − 0.258·15-s + 0.987·17-s + 0.410·19-s + 0.988·21-s + 0.859·23-s + 0.200·25-s + 0.192·27-s + 0.503·29-s + 0.837·31-s + 0.525·33-s − 0.765·35-s + 0.816·37-s − 0.799·39-s − 0.415·41-s − 1.39·43-s − 0.149·45-s − 0.398·47-s + 1.92·49-s + 0.570·51-s + 0.0427·53-s − 0.406·55-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  =  $(2,\ 8040,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.372114619$
$L(\frac12)$  $\approx$  $3.372114619$
$L(\frac{3}{2})$   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 \( 1 - 4.52T + 7T^{2} \)
11 \( 1 - 3.01T + 11T^{2} \)
13 \( 1 + 4.99T + 13T^{2} \)
17 \( 1 - 4.07T + 17T^{2} \)
19 \( 1 - 1.78T + 19T^{2} \)
23 \( 1 - 4.12T + 23T^{2} \)
29 \( 1 - 2.71T + 29T^{2} \)
31 \( 1 - 4.66T + 31T^{2} \)
37 \( 1 - 4.96T + 37T^{2} \)
41 \( 1 + 2.66T + 41T^{2} \)
43 \( 1 + 9.11T + 43T^{2} \)
47 \( 1 + 2.72T + 47T^{2} \)
53 \( 1 - 0.311T + 53T^{2} \)
59 \( 1 + 2.27T + 59T^{2} \)
61 \( 1 + 0.999T + 61T^{2} \)
71 \( 1 + 3.73T + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 + 2.72T + 79T^{2} \)
83 \( 1 + 8.69T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 - 8.55T + 97T^{2} \)
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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

−7.965957204033953654083612296107, −7.27991287696727650508002886244, −6.72144733440786819724510852247, −5.54686844988109316746701328186, −4.80509230187638534131128542221, −4.48268099171924691011748181613, −3.46834827105846367734832791169, −2.68900605842701647950875201038, −1.71135429017082406503312239783, −0.960880405052537624384991842643, 0.960880405052537624384991842643, 1.71135429017082406503312239783, 2.68900605842701647950875201038, 3.46834827105846367734832791169, 4.48268099171924691011748181613, 4.80509230187638534131128542221, 5.54686844988109316746701328186, 6.72144733440786819724510852247, 7.27991287696727650508002886244, 7.965957204033953654083612296107

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