Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 167 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3.77·5-s + 1.75·7-s + 9-s − 2.41·11-s − 1.58·13-s − 3.77·15-s + 0.811·17-s − 3.41·19-s − 1.75·21-s − 6.93·23-s + 9.26·25-s − 27-s − 0.520·29-s + 3.53·31-s + 2.41·33-s + 6.62·35-s − 9.20·37-s + 1.58·39-s − 3.31·41-s − 6.00·43-s + 3.77·45-s − 12.4·47-s − 3.92·49-s − 0.811·51-s + 12.1·53-s − 9.11·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.68·5-s + 0.662·7-s + 0.333·9-s − 0.728·11-s − 0.438·13-s − 0.975·15-s + 0.196·17-s − 0.784·19-s − 0.382·21-s − 1.44·23-s + 1.85·25-s − 0.192·27-s − 0.0966·29-s + 0.635·31-s + 0.420·33-s + 1.11·35-s − 1.51·37-s + 0.253·39-s − 0.517·41-s − 0.914·43-s + 0.563·45-s − 1.81·47-s − 0.560·49-s − 0.113·51-s + 1.66·53-s − 1.22·55-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  =  1
Selberg data  =  $(2,\ 8016,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 3.77T + 5T^{2} \)
7 \( 1 - 1.75T + 7T^{2} \)
11 \( 1 + 2.41T + 11T^{2} \)
13 \( 1 + 1.58T + 13T^{2} \)
17 \( 1 - 0.811T + 17T^{2} \)
19 \( 1 + 3.41T + 19T^{2} \)
23 \( 1 + 6.93T + 23T^{2} \)
29 \( 1 + 0.520T + 29T^{2} \)
31 \( 1 - 3.53T + 31T^{2} \)
37 \( 1 + 9.20T + 37T^{2} \)
41 \( 1 + 3.31T + 41T^{2} \)
43 \( 1 + 6.00T + 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 - 12.1T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 - 3.71T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 - 0.129T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−7.38311095962233292690446801267, −6.58151298841874019712310793719, −6.10091220687478917087719750778, −5.24547937048540050264540025332, −5.08771766893199557541076940571, −4.07831821398508420092681198935, −2.86619356586590032616469429539, −2.00531465039884007892947710431, −1.53408474268995673574896956732, 0, 1.53408474268995673574896956732, 2.00531465039884007892947710431, 2.86619356586590032616469429539, 4.07831821398508420092681198935, 5.08771766893199557541076940571, 5.24547937048540050264540025332, 6.10091220687478917087719750778, 6.58151298841874019712310793719, 7.38311095962233292690446801267

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