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 − 3.05·5-s − 2.59·7-s + 9-s − 4.72·11-s + 6.74·13-s + 3.05·15-s − 3.13·17-s − 2.68·19-s + 2.59·21-s − 6.99·23-s + 4.33·25-s − 27-s − 7.69·29-s − 4.05·31-s + 4.72·33-s + 7.94·35-s + 0.127·37-s − 6.74·39-s − 8.00·41-s + 10.7·43-s − 3.05·45-s − 1.96·47-s − 0.240·49-s + 3.13·51-s − 7.56·53-s + 14.4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.36·5-s − 0.982·7-s + 0.333·9-s − 1.42·11-s + 1.86·13-s + 0.789·15-s − 0.760·17-s − 0.616·19-s + 0.567·21-s − 1.45·23-s + 0.867·25-s − 0.192·27-s − 1.42·29-s − 0.728·31-s + 0.821·33-s + 1.34·35-s + 0.0209·37-s − 1.07·39-s − 1.24·41-s + 1.63·43-s − 0.455·45-s − 0.286·47-s − 0.0343·49-s + 0.439·51-s − 1.03·53-s + 1.94·55-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  =  $(2,\ 8016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.02368911211$
$L(\frac12)$  $\approx$  $0.02368911211$
$L(\frac{3}{2})$   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 \( 1 + 3.05T + 5T^{2} \)
7 \( 1 + 2.59T + 7T^{2} \)
11 \( 1 + 4.72T + 11T^{2} \)
13 \( 1 - 6.74T + 13T^{2} \)
17 \( 1 + 3.13T + 17T^{2} \)
19 \( 1 + 2.68T + 19T^{2} \)
23 \( 1 + 6.99T + 23T^{2} \)
29 \( 1 + 7.69T + 29T^{2} \)
31 \( 1 + 4.05T + 31T^{2} \)
37 \( 1 - 0.127T + 37T^{2} \)
41 \( 1 + 8.00T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + 1.96T + 47T^{2} \)
53 \( 1 + 7.56T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 - 2.45T + 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 - 3.69T + 71T^{2} \)
73 \( 1 - 4.93T + 73T^{2} \)
79 \( 1 + 9.15T + 79T^{2} \)
83 \( 1 + 7.82T + 83T^{2} \)
89 \( 1 - 5.37T + 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−7.78069852600378302451005268193, −7.24111836249116877908401563595, −6.24768244984903850453848177379, −5.99879276206139196022562499758, −5.02854861775492438311435109838, −4.02363460374961897088931844882, −3.77514901282565091355738574347, −2.85789341619513533705050665146, −1.66538724268454926844756238908, −0.07605181808885165553417481566, 0.07605181808885165553417481566, 1.66538724268454926844756238908, 2.85789341619513533705050665146, 3.77514901282565091355738574347, 4.02363460374961897088931844882, 5.02854861775492438311435109838, 5.99879276206139196022562499758, 6.24768244984903850453848177379, 7.24111836249116877908401563595, 7.78069852600378302451005268193

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