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 + 3.59·5-s − 4.10·7-s + 9-s − 1.62·11-s + 5.55·13-s − 3.59·15-s − 5.51·17-s + 0.688·19-s + 4.10·21-s + 4.57·23-s + 7.89·25-s − 27-s + 3.57·29-s − 7.18·31-s + 1.62·33-s − 14.7·35-s − 10.0·37-s − 5.55·39-s − 9.17·41-s − 2.89·43-s + 3.59·45-s + 5.70·47-s + 9.87·49-s + 5.51·51-s − 0.787·53-s − 5.82·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.60·5-s − 1.55·7-s + 0.333·9-s − 0.488·11-s + 1.53·13-s − 0.927·15-s − 1.33·17-s + 0.158·19-s + 0.896·21-s + 0.954·23-s + 1.57·25-s − 0.192·27-s + 0.663·29-s − 1.29·31-s + 0.282·33-s − 2.49·35-s − 1.65·37-s − 0.889·39-s − 1.43·41-s − 0.440·43-s + 0.535·45-s + 0.831·47-s + 1.41·49-s + 0.772·51-s − 0.108·53-s − 0.784·55-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 \( 1 - 3.59T + 5T^{2} \)
7 \( 1 + 4.10T + 7T^{2} \)
11 \( 1 + 1.62T + 11T^{2} \)
13 \( 1 - 5.55T + 13T^{2} \)
17 \( 1 + 5.51T + 17T^{2} \)
19 \( 1 - 0.688T + 19T^{2} \)
23 \( 1 - 4.57T + 23T^{2} \)
29 \( 1 - 3.57T + 29T^{2} \)
31 \( 1 + 7.18T + 31T^{2} \)
37 \( 1 + 10.0T + 37T^{2} \)
41 \( 1 + 9.17T + 41T^{2} \)
43 \( 1 + 2.89T + 43T^{2} \)
47 \( 1 - 5.70T + 47T^{2} \)
53 \( 1 + 0.787T + 53T^{2} \)
59 \( 1 - 2.75T + 59T^{2} \)
61 \( 1 + 1.60T + 61T^{2} \)
67 \( 1 + 7.32T + 67T^{2} \)
71 \( 1 - 6.98T + 71T^{2} \)
73 \( 1 - 0.181T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 + 3.82T + 83T^{2} \)
89 \( 1 + 6.78T + 89T^{2} \)
97 \( 1 + 7.41T + 97T^{2} \)
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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

−6.91436567942279681787701186838, −6.79269059017411627677187427501, −6.10875705887663050838141732839, −5.56524592788849152505771919842, −4.94793385817902456989358722016, −3.77347816416322846370956567002, −3.09143177073651768355307218427, −2.17295243642728938082192621630, −1.28974588478810733114371552872, 0, 1.28974588478810733114371552872, 2.17295243642728938082192621630, 3.09143177073651768355307218427, 3.77347816416322846370956567002, 4.94793385817902456989358722016, 5.56524592788849152505771919842, 6.10875705887663050838141732839, 6.79269059017411627677187427501, 6.91436567942279681787701186838

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