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 − 3.64·5-s + 1.45·7-s + 9-s + 6.42·11-s − 4.46·13-s + 3.64·15-s − 3.08·17-s + 1.23·19-s − 1.45·21-s − 1.06·23-s + 8.24·25-s − 27-s + 1.27·29-s − 4.58·31-s − 6.42·33-s − 5.28·35-s + 3.59·37-s + 4.46·39-s − 9.41·41-s − 1.44·43-s − 3.64·45-s + 3.82·47-s − 4.89·49-s + 3.08·51-s + 9.67·53-s − 23.3·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.62·5-s + 0.548·7-s + 0.333·9-s + 1.93·11-s − 1.23·13-s + 0.939·15-s − 0.748·17-s + 0.282·19-s − 0.316·21-s − 0.222·23-s + 1.64·25-s − 0.192·27-s + 0.236·29-s − 0.823·31-s − 1.11·33-s − 0.893·35-s + 0.591·37-s + 0.715·39-s − 1.46·41-s − 0.220·43-s − 0.542·45-s + 0.557·47-s − 0.698·49-s + 0.432·51-s + 1.32·53-s − 3.15·55-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 \( 1 + 3.64T + 5T^{2} \)
7 \( 1 - 1.45T + 7T^{2} \)
11 \( 1 - 6.42T + 11T^{2} \)
13 \( 1 + 4.46T + 13T^{2} \)
17 \( 1 + 3.08T + 17T^{2} \)
19 \( 1 - 1.23T + 19T^{2} \)
23 \( 1 + 1.06T + 23T^{2} \)
29 \( 1 - 1.27T + 29T^{2} \)
31 \( 1 + 4.58T + 31T^{2} \)
37 \( 1 - 3.59T + 37T^{2} \)
41 \( 1 + 9.41T + 41T^{2} \)
43 \( 1 + 1.44T + 43T^{2} \)
47 \( 1 - 3.82T + 47T^{2} \)
53 \( 1 - 9.67T + 53T^{2} \)
59 \( 1 + 7.68T + 59T^{2} \)
61 \( 1 - 6.50T + 61T^{2} \)
67 \( 1 - 7.73T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 - 2.25T + 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 + 2.64T + 83T^{2} \)
89 \( 1 - 3.71T + 89T^{2} \)
97 \( 1 - 5.92T + 97T^{2} \)
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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

−7.28692987295149226255196599328, −6.99733398577002535221752510434, −6.27054580902648294743053090222, −5.18592349238377905427374172417, −4.57091180373200578781505713364, −4.01998128925268829988724156588, −3.40472229413822825291370400963, −2.11336000904444405780114555196, −1.04757541224054261584707964823, 0, 1.04757541224054261584707964823, 2.11336000904444405780114555196, 3.40472229413822825291370400963, 4.01998128925268829988724156588, 4.57091180373200578781505713364, 5.18592349238377905427374172417, 6.27054580902648294743053090222, 6.99733398577002535221752510434, 7.28692987295149226255196599328

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