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 − 0.274·7-s + 9-s − 4.75·11-s − 2.11·13-s − 15-s + 3.21·17-s − 1.31·19-s − 0.274·21-s − 4.76·23-s + 25-s + 27-s − 6.42·29-s + 3.56·31-s − 4.75·33-s + 0.274·35-s + 9.24·37-s − 2.11·39-s − 1.56·41-s + 2.64·43-s − 45-s + 7.55·47-s − 6.92·49-s + 3.21·51-s − 12.6·53-s + 4.75·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.103·7-s + 0.333·9-s − 1.43·11-s − 0.587·13-s − 0.258·15-s + 0.779·17-s − 0.302·19-s − 0.0598·21-s − 0.994·23-s + 0.200·25-s + 0.192·27-s − 1.19·29-s + 0.639·31-s − 0.827·33-s + 0.0463·35-s + 1.51·37-s − 0.339·39-s − 0.244·41-s + 0.404·43-s − 0.149·45-s + 1.10·47-s − 0.989·49-s + 0.450·51-s − 1.73·53-s + 0.640·55-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  =  $(2,\ 8040,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.610602413$
$L(\frac12)$  $\approx$  $1.610602413$
$L(\frac{3}{2})$   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 \( 1 + 0.274T + 7T^{2} \)
11 \( 1 + 4.75T + 11T^{2} \)
13 \( 1 + 2.11T + 13T^{2} \)
17 \( 1 - 3.21T + 17T^{2} \)
19 \( 1 + 1.31T + 19T^{2} \)
23 \( 1 + 4.76T + 23T^{2} \)
29 \( 1 + 6.42T + 29T^{2} \)
31 \( 1 - 3.56T + 31T^{2} \)
37 \( 1 - 9.24T + 37T^{2} \)
41 \( 1 + 1.56T + 41T^{2} \)
43 \( 1 - 2.64T + 43T^{2} \)
47 \( 1 - 7.55T + 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 - 5.61T + 61T^{2} \)
71 \( 1 - 5.03T + 71T^{2} \)
73 \( 1 - 0.416T + 73T^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + 8.57T + 83T^{2} \)
89 \( 1 - 6.86T + 89T^{2} \)
97 \( 1 + 11.2T + 97T^{2} \)
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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

−7.83551250443032902505892552517, −7.46557446395586193514394019981, −6.50341709124031513203693346872, −5.68706066366431565166358202931, −4.99777877561630857729807129349, −4.22409000479842212363277377871, −3.44995415818261639306327940672, −2.66551215691499248762232404322, −2.00665248696187530211995344546, −0.58280721254064627998106802362, 0.58280721254064627998106802362, 2.00665248696187530211995344546, 2.66551215691499248762232404322, 3.44995415818261639306327940672, 4.22409000479842212363277377871, 4.99777877561630857729807129349, 5.68706066366431565166358202931, 6.50341709124031513203693346872, 7.46557446395586193514394019981, 7.83551250443032902505892552517

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