Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 167 $
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 − 3.60·5-s + 4.85·7-s + 9-s − 0.783·11-s − 2.28·13-s + 3.60·15-s − 3.64·17-s + 1.90·19-s − 4.85·21-s − 0.781·23-s + 7.96·25-s − 27-s − 5.79·29-s + 5.38·31-s + 0.783·33-s − 17.4·35-s − 7.39·37-s + 2.28·39-s + 1.19·41-s − 2.44·43-s − 3.60·45-s + 0.0708·47-s + 16.5·49-s + 3.64·51-s + 2.21·53-s + 2.82·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.61·5-s + 1.83·7-s + 0.333·9-s − 0.236·11-s − 0.634·13-s + 0.929·15-s − 0.883·17-s + 0.436·19-s − 1.06·21-s − 0.163·23-s + 1.59·25-s − 0.192·27-s − 1.07·29-s + 0.967·31-s + 0.136·33-s − 2.95·35-s − 1.21·37-s + 0.366·39-s + 0.185·41-s − 0.373·43-s − 0.536·45-s + 0.0103·47-s + 2.37·49-s + 0.510·51-s + 0.303·53-s + 0.380·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8016} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.057673448$
$L(\frac12)$  $\approx$  $1.057673448$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
167 \( 1 - T \)
good5 \( 1 + 3.60T + 5T^{2} \)
7 \( 1 - 4.85T + 7T^{2} \)
11 \( 1 + 0.783T + 11T^{2} \)
13 \( 1 + 2.28T + 13T^{2} \)
17 \( 1 + 3.64T + 17T^{2} \)
19 \( 1 - 1.90T + 19T^{2} \)
23 \( 1 + 0.781T + 23T^{2} \)
29 \( 1 + 5.79T + 29T^{2} \)
31 \( 1 - 5.38T + 31T^{2} \)
37 \( 1 + 7.39T + 37T^{2} \)
41 \( 1 - 1.19T + 41T^{2} \)
43 \( 1 + 2.44T + 43T^{2} \)
47 \( 1 - 0.0708T + 47T^{2} \)
53 \( 1 - 2.21T + 53T^{2} \)
59 \( 1 - 3.50T + 59T^{2} \)
61 \( 1 - 15.0T + 61T^{2} \)
67 \( 1 + 2.58T + 67T^{2} \)
71 \( 1 - 6.82T + 71T^{2} \)
73 \( 1 - 8.47T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + 16.5T + 83T^{2} \)
89 \( 1 - 0.933T + 89T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−7.932902825312491378034313527323, −7.20143077517558540576651372661, −6.73272150371579816337578386397, −5.32383848489695701527647375895, −5.13497603562527686312309139465, −4.24765866074212550810657400244, −3.89481005046153804601178212773, −2.61487980097738603090007833990, −1.64352990704299605335983856968, −0.53494854254086693540266079804, 0.53494854254086693540266079804, 1.64352990704299605335983856968, 2.61487980097738603090007833990, 3.89481005046153804601178212773, 4.24765866074212550810657400244, 5.13497603562527686312309139465, 5.32383848489695701527647375895, 6.73272150371579816337578386397, 7.20143077517558540576651372661, 7.932902825312491378034313527323

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