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 + 0.284·5-s + 3.90·7-s + 9-s − 5.93·11-s + 5.62·13-s − 0.284·15-s − 0.211·17-s + 1.66·19-s − 3.90·21-s + 2.16·23-s − 4.91·25-s − 27-s + 4.18·29-s + 2.41·31-s + 5.93·33-s + 1.10·35-s + 0.0651·37-s − 5.62·39-s + 6.23·41-s − 2.49·43-s + 0.284·45-s − 5.97·47-s + 8.26·49-s + 0.211·51-s + 4.43·53-s − 1.68·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.127·5-s + 1.47·7-s + 0.333·9-s − 1.79·11-s + 1.55·13-s − 0.0733·15-s − 0.0513·17-s + 0.381·19-s − 0.852·21-s + 0.451·23-s − 0.983·25-s − 0.192·27-s + 0.777·29-s + 0.433·31-s + 1.03·33-s + 0.187·35-s + 0.0107·37-s − 0.900·39-s + 0.973·41-s − 0.379·43-s + 0.0423·45-s − 0.871·47-s + 1.18·49-s + 0.0296·51-s + 0.609·53-s − 0.227·55-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  =  $(2,\ 8016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.047222902$
$L(\frac12)$  $\approx$  $2.047222902$
$L(\frac{3}{2})$   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 \( 1 - 0.284T + 5T^{2} \)
7 \( 1 - 3.90T + 7T^{2} \)
11 \( 1 + 5.93T + 11T^{2} \)
13 \( 1 - 5.62T + 13T^{2} \)
17 \( 1 + 0.211T + 17T^{2} \)
19 \( 1 - 1.66T + 19T^{2} \)
23 \( 1 - 2.16T + 23T^{2} \)
29 \( 1 - 4.18T + 29T^{2} \)
31 \( 1 - 2.41T + 31T^{2} \)
37 \( 1 - 0.0651T + 37T^{2} \)
41 \( 1 - 6.23T + 41T^{2} \)
43 \( 1 + 2.49T + 43T^{2} \)
47 \( 1 + 5.97T + 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 - 3.03T + 59T^{2} \)
61 \( 1 + 5.18T + 61T^{2} \)
67 \( 1 - 5.08T + 67T^{2} \)
71 \( 1 + 6.06T + 71T^{2} \)
73 \( 1 - 8.31T + 73T^{2} \)
79 \( 1 - 2.87T + 79T^{2} \)
83 \( 1 - 5.50T + 83T^{2} \)
89 \( 1 - 0.660T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−8.032907854615204341709322122197, −7.25123076511011952869125509945, −6.28769241401949815932927609794, −5.66191425830925772843781761454, −5.07167210503813976997706473216, −4.53657291950723545230548380293, −3.57739137548422459244697102600, −2.55023445099100822854258413557, −1.66952229967608476220811418001, −0.76361153578716861587058377981, 0.76361153578716861587058377981, 1.66952229967608476220811418001, 2.55023445099100822854258413557, 3.57739137548422459244697102600, 4.53657291950723545230548380293, 5.07167210503813976997706473216, 5.66191425830925772843781761454, 6.28769241401949815932927609794, 7.25123076511011952869125509945, 8.032907854615204341709322122197

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