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 − 0.716·5-s + 3.33·7-s + 9-s + 1.47·11-s − 1.07·13-s + 0.716·15-s + 4.68·17-s + 1.60·19-s − 3.33·21-s − 5.96·23-s − 4.48·25-s − 27-s + 4.73·29-s − 8.82·31-s − 1.47·33-s − 2.38·35-s − 10.4·37-s + 1.07·39-s − 2.92·41-s + 8.38·43-s − 0.716·45-s − 6.24·47-s + 4.08·49-s − 4.68·51-s − 9.50·53-s − 1.05·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.320·5-s + 1.25·7-s + 0.333·9-s + 0.443·11-s − 0.297·13-s + 0.184·15-s + 1.13·17-s + 0.367·19-s − 0.726·21-s − 1.24·23-s − 0.897·25-s − 0.192·27-s + 0.878·29-s − 1.58·31-s − 0.256·33-s − 0.403·35-s − 1.71·37-s + 0.171·39-s − 0.456·41-s + 1.27·43-s − 0.106·45-s − 0.910·47-s + 0.584·49-s − 0.656·51-s − 1.30·53-s − 0.142·55-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 \( 1 + 0.716T + 5T^{2} \)
7 \( 1 - 3.33T + 7T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
13 \( 1 + 1.07T + 13T^{2} \)
17 \( 1 - 4.68T + 17T^{2} \)
19 \( 1 - 1.60T + 19T^{2} \)
23 \( 1 + 5.96T + 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 + 8.82T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 + 2.92T + 41T^{2} \)
43 \( 1 - 8.38T + 43T^{2} \)
47 \( 1 + 6.24T + 47T^{2} \)
53 \( 1 + 9.50T + 53T^{2} \)
59 \( 1 - 4.74T + 59T^{2} \)
61 \( 1 - 7.09T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 7.30T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + 1.24T + 83T^{2} \)
89 \( 1 + 7.67T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−7.55092806857670817013357354285, −6.91244300027336891230053410241, −5.89634961424951720094370122977, −5.45479031701089754173437666389, −4.70521182929408215200307254535, −4.02382986497616034886522159544, −3.25471101070498857343349896396, −1.92985697298117670519760484559, −1.34899557065709381453012415346, 0, 1.34899557065709381453012415346, 1.92985697298117670519760484559, 3.25471101070498857343349896396, 4.02382986497616034886522159544, 4.70521182929408215200307254535, 5.45479031701089754173437666389, 5.89634961424951720094370122977, 6.91244300027336891230053410241, 7.55092806857670817013357354285

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