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 + 0.670·7-s + 9-s + 3.78·11-s + 4.11·13-s − 15-s + 1.83·17-s + 5.73·19-s + 0.670·21-s + 7.13·23-s + 25-s + 27-s − 1.82·29-s − 5.12·31-s + 3.78·33-s − 0.670·35-s + 2.09·37-s + 4.11·39-s + 7.12·41-s − 3.02·43-s − 45-s − 5.47·47-s − 6.55·49-s + 1.83·51-s + 6.86·53-s − 3.78·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.253·7-s + 0.333·9-s + 1.14·11-s + 1.14·13-s − 0.258·15-s + 0.446·17-s + 1.31·19-s + 0.146·21-s + 1.48·23-s + 0.200·25-s + 0.192·27-s − 0.338·29-s − 0.919·31-s + 0.658·33-s − 0.113·35-s + 0.343·37-s + 0.658·39-s + 1.11·41-s − 0.461·43-s − 0.149·45-s − 0.797·47-s − 0.935·49-s + 0.257·51-s + 0.942·53-s − 0.510·55-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  =  $(2,\ 8040,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.264714345$
$L(\frac12)$  $\approx$  $3.264714345$
$L(\frac{3}{2})$   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 \( 1 - 0.670T + 7T^{2} \)
11 \( 1 - 3.78T + 11T^{2} \)
13 \( 1 - 4.11T + 13T^{2} \)
17 \( 1 - 1.83T + 17T^{2} \)
19 \( 1 - 5.73T + 19T^{2} \)
23 \( 1 - 7.13T + 23T^{2} \)
29 \( 1 + 1.82T + 29T^{2} \)
31 \( 1 + 5.12T + 31T^{2} \)
37 \( 1 - 2.09T + 37T^{2} \)
41 \( 1 - 7.12T + 41T^{2} \)
43 \( 1 + 3.02T + 43T^{2} \)
47 \( 1 + 5.47T + 47T^{2} \)
53 \( 1 - 6.86T + 53T^{2} \)
59 \( 1 - 1.63T + 59T^{2} \)
61 \( 1 - 5.50T + 61T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 7.01T + 79T^{2} \)
83 \( 1 + 5.15T + 83T^{2} \)
89 \( 1 - 5.63T + 89T^{2} \)
97 \( 1 + 9.80T + 97T^{2} \)
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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

−7.78397351098181048072668861713, −7.25528103838298819277780243592, −6.58870059276422346169786597266, −5.74208406811642236235587336991, −4.98861992099895170164888636551, −4.07573056471976987794517691257, −3.51913414037541566306430749201, −2.89761550669903835452931355758, −1.57131317624822094289778061884, −0.983596852258812971532847292287, 0.983596852258812971532847292287, 1.57131317624822094289778061884, 2.89761550669903835452931355758, 3.51913414037541566306430749201, 4.07573056471976987794517691257, 4.98861992099895170164888636551, 5.74208406811642236235587336991, 6.58870059276422346169786597266, 7.25528103838298819277780243592, 7.78397351098181048072668861713

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