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 − 1.80·5-s + 4.10·7-s + 9-s + 2.59·11-s − 3.46·13-s − 1.80·15-s + 1.68·17-s + 2.49·19-s + 4.10·21-s + 8.91·23-s − 1.74·25-s + 27-s − 4.42·29-s + 2.95·31-s + 2.59·33-s − 7.40·35-s + 0.918·37-s − 3.46·39-s − 1.83·41-s − 0.850·43-s − 1.80·45-s + 6.74·47-s + 9.88·49-s + 1.68·51-s + 4.65·53-s − 4.67·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.806·5-s + 1.55·7-s + 0.333·9-s + 0.781·11-s − 0.961·13-s − 0.465·15-s + 0.408·17-s + 0.573·19-s + 0.896·21-s + 1.85·23-s − 0.349·25-s + 0.192·27-s − 0.821·29-s + 0.529·31-s + 0.451·33-s − 1.25·35-s + 0.150·37-s − 0.555·39-s − 0.287·41-s − 0.129·43-s − 0.268·45-s + 0.983·47-s + 1.41·49-s + 0.235·51-s + 0.639·53-s − 0.630·55-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  =  \((2,\ 8016,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.986872389\)
\(L(\frac12)\)  \(\approx\)  \(2.986872389\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
167 \( 1 + T \)
good5 \( 1 + 1.80T + 5T^{2} \)
7 \( 1 - 4.10T + 7T^{2} \)
11 \( 1 - 2.59T + 11T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 - 1.68T + 17T^{2} \)
19 \( 1 - 2.49T + 19T^{2} \)
23 \( 1 - 8.91T + 23T^{2} \)
29 \( 1 + 4.42T + 29T^{2} \)
31 \( 1 - 2.95T + 31T^{2} \)
37 \( 1 - 0.918T + 37T^{2} \)
41 \( 1 + 1.83T + 41T^{2} \)
43 \( 1 + 0.850T + 43T^{2} \)
47 \( 1 - 6.74T + 47T^{2} \)
53 \( 1 - 4.65T + 53T^{2} \)
59 \( 1 + 1.23T + 59T^{2} \)
61 \( 1 + 7.97T + 61T^{2} \)
67 \( 1 - 1.65T + 67T^{2} \)
71 \( 1 - 6.24T + 71T^{2} \)
73 \( 1 + 4.76T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 3.69T + 83T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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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

−7.69822243112553543900798311369, −7.45579150973741705656542463726, −6.72746457485990525995970968682, −5.51977582631968101610209195117, −4.91480492775784136312209740454, −4.30120205972611558997069408020, −3.56541947932289919452942548475, −2.69610479640718123079042351509, −1.73478722833450916906733578568, −0.885872249558148497873950069949, 0.885872249558148497873950069949, 1.73478722833450916906733578568, 2.69610479640718123079042351509, 3.56541947932289919452942548475, 4.30120205972611558997069408020, 4.91480492775784136312209740454, 5.51977582631968101610209195117, 6.72746457485990525995970968682, 7.45579150973741705656542463726, 7.69822243112553543900798311369

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