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 − 0.0932·5-s − 3.86·7-s + 9-s − 4.61·11-s − 0.601·13-s − 0.0932·15-s − 0.832·17-s − 3.67·19-s − 3.86·21-s − 6.30·23-s − 4.99·25-s + 27-s + 4.62·29-s − 4.74·31-s − 4.61·33-s + 0.360·35-s + 5.70·37-s − 0.601·39-s − 4.02·41-s + 3.85·43-s − 0.0932·45-s − 2.09·47-s + 7.91·49-s − 0.832·51-s + 7.76·53-s + 0.430·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.0417·5-s − 1.45·7-s + 0.333·9-s − 1.39·11-s − 0.166·13-s − 0.0240·15-s − 0.201·17-s − 0.843·19-s − 0.842·21-s − 1.31·23-s − 0.998·25-s + 0.192·27-s + 0.858·29-s − 0.851·31-s − 0.803·33-s + 0.0608·35-s + 0.937·37-s − 0.0963·39-s − 0.627·41-s + 0.587·43-s − 0.0139·45-s − 0.305·47-s + 1.13·49-s − 0.116·51-s + 1.06·53-s + 0.0580·55-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  =  \((2,\ 8016,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.029424698\)
\(L(\frac12)\)  \(\approx\)  \(1.029424698\)
\(L(\frac{3}{2})\)   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 \( 1 + 0.0932T + 5T^{2} \)
7 \( 1 + 3.86T + 7T^{2} \)
11 \( 1 + 4.61T + 11T^{2} \)
13 \( 1 + 0.601T + 13T^{2} \)
17 \( 1 + 0.832T + 17T^{2} \)
19 \( 1 + 3.67T + 19T^{2} \)
23 \( 1 + 6.30T + 23T^{2} \)
29 \( 1 - 4.62T + 29T^{2} \)
31 \( 1 + 4.74T + 31T^{2} \)
37 \( 1 - 5.70T + 37T^{2} \)
41 \( 1 + 4.02T + 41T^{2} \)
43 \( 1 - 3.85T + 43T^{2} \)
47 \( 1 + 2.09T + 47T^{2} \)
53 \( 1 - 7.76T + 53T^{2} \)
59 \( 1 - 4.38T + 59T^{2} \)
61 \( 1 - 1.61T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 - 2.17T + 73T^{2} \)
79 \( 1 - 8.12T + 79T^{2} \)
83 \( 1 - 2.46T + 83T^{2} \)
89 \( 1 - 4.23T + 89T^{2} \)
97 \( 1 - 3.07T + 97T^{2} \)
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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

−8.024783529472893652431808151779, −7.14267751529092346503721624454, −6.48564352532684289837101186256, −5.85443589940186763298441598138, −5.06096151147215322593228054769, −4.03581442320935693473582513985, −3.54750590522834890252831900185, −2.54484615811875143001192751733, −2.17372481983245762869181118046, −0.44973376264797522626689594614, 0.44973376264797522626689594614, 2.17372481983245762869181118046, 2.54484615811875143001192751733, 3.54750590522834890252831900185, 4.03581442320935693473582513985, 5.06096151147215322593228054769, 5.85443589940186763298441598138, 6.48564352532684289837101186256, 7.14267751529092346503721624454, 8.024783529472893652431808151779

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