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 − 2.09·5-s − 2.97·7-s + 9-s + 0.138·11-s + 5.06·13-s + 2.09·15-s + 0.299·17-s − 0.432·19-s + 2.97·21-s − 5.72·23-s − 0.590·25-s − 27-s − 0.718·29-s + 4.27·31-s − 0.138·33-s + 6.24·35-s − 3.92·37-s − 5.06·39-s + 7.57·41-s − 12.3·43-s − 2.09·45-s + 9.22·47-s + 1.84·49-s − 0.299·51-s + 1.26·53-s − 0.290·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.939·5-s − 1.12·7-s + 0.333·9-s + 0.0416·11-s + 1.40·13-s + 0.542·15-s + 0.0725·17-s − 0.0991·19-s + 0.649·21-s − 1.19·23-s − 0.118·25-s − 0.192·27-s − 0.133·29-s + 0.767·31-s − 0.0240·33-s + 1.05·35-s − 0.644·37-s − 0.810·39-s + 1.18·41-s − 1.88·43-s − 0.313·45-s + 1.34·47-s + 0.263·49-s − 0.0418·51-s + 0.173·53-s − 0.0391·55-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 \( 1 + 2.09T + 5T^{2} \)
7 \( 1 + 2.97T + 7T^{2} \)
11 \( 1 - 0.138T + 11T^{2} \)
13 \( 1 - 5.06T + 13T^{2} \)
17 \( 1 - 0.299T + 17T^{2} \)
19 \( 1 + 0.432T + 19T^{2} \)
23 \( 1 + 5.72T + 23T^{2} \)
29 \( 1 + 0.718T + 29T^{2} \)
31 \( 1 - 4.27T + 31T^{2} \)
37 \( 1 + 3.92T + 37T^{2} \)
41 \( 1 - 7.57T + 41T^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
47 \( 1 - 9.22T + 47T^{2} \)
53 \( 1 - 1.26T + 53T^{2} \)
59 \( 1 + 3.01T + 59T^{2} \)
61 \( 1 - 2.98T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 - 7.07T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 - 7.54T + 97T^{2} \)
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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

−7.45325489529972498270974659283, −6.63373219508609409925086038513, −6.21705726257624722230627124402, −5.55513578081057818827175562465, −4.55189049864615075364189038279, −3.70921900717218799495307710798, −3.50689841750590751728970377936, −2.23198996636842962619411684562, −0.970269029790897361051313473636, 0, 0.970269029790897361051313473636, 2.23198996636842962619411684562, 3.50689841750590751728970377936, 3.70921900717218799495307710798, 4.55189049864615075364189038279, 5.55513578081057818827175562465, 6.21705726257624722230627124402, 6.63373219508609409925086038513, 7.45325489529972498270974659283

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