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 + 1.25·5-s + 4.55·7-s + 9-s + 5.23·11-s − 1.93·13-s − 1.25·15-s + 7.59·17-s + 4.42·19-s − 4.55·21-s − 6.61·23-s − 3.42·25-s − 27-s + 3.73·29-s + 1.27·31-s − 5.23·33-s + 5.72·35-s + 6.87·37-s + 1.93·39-s + 8.71·41-s − 5.25·43-s + 1.25·45-s + 6.55·47-s + 13.7·49-s − 7.59·51-s + 3.93·53-s + 6.57·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.562·5-s + 1.72·7-s + 0.333·9-s + 1.57·11-s − 0.536·13-s − 0.324·15-s + 1.84·17-s + 1.01·19-s − 0.994·21-s − 1.37·23-s − 0.684·25-s − 0.192·27-s + 0.693·29-s + 0.228·31-s − 0.911·33-s + 0.967·35-s + 1.12·37-s + 0.309·39-s + 1.36·41-s − 0.801·43-s + 0.187·45-s + 0.956·47-s + 1.96·49-s − 1.06·51-s + 0.541·53-s + 0.887·55-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  =  $(2,\ 8016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.089625486$
$L(\frac12)$  $\approx$  $3.089625486$
$L(\frac{3}{2})$   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 \( 1 - 1.25T + 5T^{2} \)
7 \( 1 - 4.55T + 7T^{2} \)
11 \( 1 - 5.23T + 11T^{2} \)
13 \( 1 + 1.93T + 13T^{2} \)
17 \( 1 - 7.59T + 17T^{2} \)
19 \( 1 - 4.42T + 19T^{2} \)
23 \( 1 + 6.61T + 23T^{2} \)
29 \( 1 - 3.73T + 29T^{2} \)
31 \( 1 - 1.27T + 31T^{2} \)
37 \( 1 - 6.87T + 37T^{2} \)
41 \( 1 - 8.71T + 41T^{2} \)
43 \( 1 + 5.25T + 43T^{2} \)
47 \( 1 - 6.55T + 47T^{2} \)
53 \( 1 - 3.93T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 + 0.282T + 61T^{2} \)
67 \( 1 - 8.25T + 67T^{2} \)
71 \( 1 - 1.66T + 71T^{2} \)
73 \( 1 + 3.89T + 73T^{2} \)
79 \( 1 + 6.21T + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 + 6.93T + 89T^{2} \)
97 \( 1 + 3.11T + 97T^{2} \)
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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

−7.66235933844664403446869451286, −7.37016514997880932859308690721, −6.16711922434182262978075686903, −5.81847990393578782625349332065, −5.09559686488064618034723459547, −4.38544475097262349176748740082, −3.73175290795900905548652163434, −2.47827136972535191418501082054, −1.46813786678845969533585755582, −1.07434519172689328599194454799, 1.07434519172689328599194454799, 1.46813786678845969533585755582, 2.47827136972535191418501082054, 3.73175290795900905548652163434, 4.38544475097262349176748740082, 5.09559686488064618034723459547, 5.81847990393578782625349332065, 6.16711922434182262978075686903, 7.37016514997880932859308690721, 7.66235933844664403446869451286

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