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 − 0.619·5-s − 0.954·7-s + 9-s − 6.25·11-s + 2.92·13-s + 0.619·15-s + 5.33·17-s − 2.94·19-s + 0.954·21-s + 2.33·23-s − 4.61·25-s − 27-s + 4.47·29-s + 3.45·31-s + 6.25·33-s + 0.590·35-s + 4.26·37-s − 2.92·39-s − 10.8·41-s + 1.32·43-s − 0.619·45-s − 3.37·47-s − 6.08·49-s − 5.33·51-s + 5.72·53-s + 3.87·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.276·5-s − 0.360·7-s + 0.333·9-s − 1.88·11-s + 0.811·13-s + 0.159·15-s + 1.29·17-s − 0.675·19-s + 0.208·21-s + 0.487·23-s − 0.923·25-s − 0.192·27-s + 0.830·29-s + 0.620·31-s + 1.08·33-s + 0.0998·35-s + 0.701·37-s − 0.468·39-s − 1.68·41-s + 0.201·43-s − 0.0923·45-s − 0.491·47-s − 0.869·49-s − 0.747·51-s + 0.786·53-s + 0.522·55-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 \( 1 + 0.619T + 5T^{2} \)
7 \( 1 + 0.954T + 7T^{2} \)
11 \( 1 + 6.25T + 11T^{2} \)
13 \( 1 - 2.92T + 13T^{2} \)
17 \( 1 - 5.33T + 17T^{2} \)
19 \( 1 + 2.94T + 19T^{2} \)
23 \( 1 - 2.33T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 - 3.45T + 31T^{2} \)
37 \( 1 - 4.26T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 1.32T + 43T^{2} \)
47 \( 1 + 3.37T + 47T^{2} \)
53 \( 1 - 5.72T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 + 0.604T + 67T^{2} \)
71 \( 1 - 9.74T + 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 + 6.74T + 79T^{2} \)
83 \( 1 + 0.00272T + 83T^{2} \)
89 \( 1 - 9.85T + 89T^{2} \)
97 \( 1 + 8.92T + 97T^{2} \)
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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

−7.52241315505555010147835703354, −6.75093056902722617354143749404, −6.03905805317501716950527504756, −5.39308109128359208982701155503, −4.85317902733797276581362235328, −3.87288614815368594735217358584, −3.13985766846140267675086321918, −2.29925402414912427514326742273, −1.05015736327502766854780313881, 0, 1.05015736327502766854780313881, 2.29925402414912427514326742273, 3.13985766846140267675086321918, 3.87288614815368594735217358584, 4.85317902733797276581362235328, 5.39308109128359208982701155503, 6.03905805317501716950527504756, 6.75093056902722617354143749404, 7.52241315505555010147835703354

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