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 − 4.02·5-s − 0.910·7-s + 9-s + 5.64·11-s + 6.43·13-s + 4.02·15-s + 3.30·17-s − 1.28·19-s + 0.910·21-s + 9.24·23-s + 11.2·25-s − 27-s + 2.90·29-s + 10.6·31-s − 5.64·33-s + 3.66·35-s + 3.18·37-s − 6.43·39-s + 1.79·41-s − 2.55·43-s − 4.02·45-s − 1.91·47-s − 6.17·49-s − 3.30·51-s − 6.55·53-s − 22.7·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.80·5-s − 0.343·7-s + 0.333·9-s + 1.70·11-s + 1.78·13-s + 1.03·15-s + 0.802·17-s − 0.295·19-s + 0.198·21-s + 1.92·23-s + 2.24·25-s − 0.192·27-s + 0.540·29-s + 1.90·31-s − 0.983·33-s + 0.619·35-s + 0.524·37-s − 1.03·39-s + 0.279·41-s − 0.390·43-s − 0.600·45-s − 0.279·47-s − 0.881·49-s − 0.463·51-s − 0.900·53-s − 3.06·55-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  =  $(2,\ 8016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.637850228$
$L(\frac12)$  $\approx$  $1.637850228$
$L(\frac{3}{2})$   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 \( 1 + 4.02T + 5T^{2} \)
7 \( 1 + 0.910T + 7T^{2} \)
11 \( 1 - 5.64T + 11T^{2} \)
13 \( 1 - 6.43T + 13T^{2} \)
17 \( 1 - 3.30T + 17T^{2} \)
19 \( 1 + 1.28T + 19T^{2} \)
23 \( 1 - 9.24T + 23T^{2} \)
29 \( 1 - 2.90T + 29T^{2} \)
31 \( 1 - 10.6T + 31T^{2} \)
37 \( 1 - 3.18T + 37T^{2} \)
41 \( 1 - 1.79T + 41T^{2} \)
43 \( 1 + 2.55T + 43T^{2} \)
47 \( 1 + 1.91T + 47T^{2} \)
53 \( 1 + 6.55T + 53T^{2} \)
59 \( 1 - 2.97T + 59T^{2} \)
61 \( 1 - 8.34T + 61T^{2} \)
67 \( 1 + 4.23T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 1.27T + 73T^{2} \)
79 \( 1 - 1.79T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 5.45T + 89T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−7.84488278688287963282301900008, −6.98751162528583976527340236093, −6.54078104932147231316975770000, −5.98336344814664128855626780458, −4.74103373230765696226932546421, −4.32251156690414185183710414182, −3.47519622687774994351477622607, −3.17423856880739319758353085848, −1.21320863789813250333520983365, −0.822973061741726542234016974706, 0.822973061741726542234016974706, 1.21320863789813250333520983365, 3.17423856880739319758353085848, 3.47519622687774994351477622607, 4.32251156690414185183710414182, 4.74103373230765696226932546421, 5.98336344814664128855626780458, 6.54078104932147231316975770000, 6.98751162528583976527340236093, 7.84488278688287963282301900008

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