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 + 0.649·5-s − 4.19·7-s + 9-s − 1.95·11-s + 3.07·13-s − 0.649·15-s + 6.85·17-s − 8.04·19-s + 4.19·21-s + 6.50·23-s − 4.57·25-s − 27-s − 2.52·29-s − 4.14·31-s + 1.95·33-s − 2.72·35-s − 4.94·37-s − 3.07·39-s + 2.05·41-s − 4.35·43-s + 0.649·45-s + 11.2·47-s + 10.6·49-s − 6.85·51-s − 1.69·53-s − 1.26·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.290·5-s − 1.58·7-s + 0.333·9-s − 0.588·11-s + 0.852·13-s − 0.167·15-s + 1.66·17-s − 1.84·19-s + 0.915·21-s + 1.35·23-s − 0.915·25-s − 0.192·27-s − 0.468·29-s − 0.745·31-s + 0.339·33-s − 0.460·35-s − 0.813·37-s − 0.492·39-s + 0.321·41-s − 0.664·43-s + 0.0967·45-s + 1.63·47-s + 1.51·49-s − 0.959·51-s − 0.233·53-s − 0.170·55-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  =  $(2,\ 8016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9741133919$
$L(\frac12)$  $\approx$  $0.9741133919$
$L(\frac{3}{2})$   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 \( 1 - 0.649T + 5T^{2} \)
7 \( 1 + 4.19T + 7T^{2} \)
11 \( 1 + 1.95T + 11T^{2} \)
13 \( 1 - 3.07T + 13T^{2} \)
17 \( 1 - 6.85T + 17T^{2} \)
19 \( 1 + 8.04T + 19T^{2} \)
23 \( 1 - 6.50T + 23T^{2} \)
29 \( 1 + 2.52T + 29T^{2} \)
31 \( 1 + 4.14T + 31T^{2} \)
37 \( 1 + 4.94T + 37T^{2} \)
41 \( 1 - 2.05T + 41T^{2} \)
43 \( 1 + 4.35T + 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 + 1.69T + 53T^{2} \)
59 \( 1 - 7.31T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 - 6.97T + 67T^{2} \)
71 \( 1 + 2.84T + 71T^{2} \)
73 \( 1 + 7.78T + 73T^{2} \)
79 \( 1 - 3.29T + 79T^{2} \)
83 \( 1 + 1.99T + 83T^{2} \)
89 \( 1 + 15.9T + 89T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−7.68705606097783967513991986685, −7.00993758304998797886014112562, −6.35697562886732724440662755805, −5.78451157732877450039653251447, −5.33757466517874813508248264770, −4.16542188980766773253241244100, −3.51778802408783642814274206629, −2.79918768738770842869208288892, −1.68943499919139246665170233634, −0.49731862659161196021048131806, 0.49731862659161196021048131806, 1.68943499919139246665170233634, 2.79918768738770842869208288892, 3.51778802408783642814274206629, 4.16542188980766773253241244100, 5.33757466517874813508248264770, 5.78451157732877450039653251447, 6.35697562886732724440662755805, 7.00993758304998797886014112562, 7.68705606097783967513991986685

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