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 + 3.17·5-s − 0.651·7-s + 9-s − 2.35·11-s + 5.77·13-s + 3.17·15-s − 0.736·17-s + 3.04·19-s − 0.651·21-s + 1.45·23-s + 5.10·25-s + 27-s + 3.59·29-s − 5.65·31-s − 2.35·33-s − 2.07·35-s + 7.15·37-s + 5.77·39-s + 8.85·41-s + 4.96·43-s + 3.17·45-s − 3.18·47-s − 6.57·49-s − 0.736·51-s + 1.50·53-s − 7.48·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.42·5-s − 0.246·7-s + 0.333·9-s − 0.710·11-s + 1.60·13-s + 0.820·15-s − 0.178·17-s + 0.699·19-s − 0.142·21-s + 0.303·23-s + 1.02·25-s + 0.192·27-s + 0.667·29-s − 1.01·31-s − 0.410·33-s − 0.350·35-s + 1.17·37-s + 0.925·39-s + 1.38·41-s + 0.757·43-s + 0.473·45-s − 0.463·47-s − 0.939·49-s − 0.103·51-s + 0.206·53-s − 1.00·55-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  =  \((2,\ 8016,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.802787470\)
\(L(\frac12)\)  \(\approx\)  \(3.802787470\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
167 \( 1 + T \)
good5 \( 1 - 3.17T + 5T^{2} \)
7 \( 1 + 0.651T + 7T^{2} \)
11 \( 1 + 2.35T + 11T^{2} \)
13 \( 1 - 5.77T + 13T^{2} \)
17 \( 1 + 0.736T + 17T^{2} \)
19 \( 1 - 3.04T + 19T^{2} \)
23 \( 1 - 1.45T + 23T^{2} \)
29 \( 1 - 3.59T + 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 - 7.15T + 37T^{2} \)
41 \( 1 - 8.85T + 41T^{2} \)
43 \( 1 - 4.96T + 43T^{2} \)
47 \( 1 + 3.18T + 47T^{2} \)
53 \( 1 - 1.50T + 53T^{2} \)
59 \( 1 + 3.09T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 - 6.06T + 67T^{2} \)
71 \( 1 - 6.79T + 71T^{2} \)
73 \( 1 + 1.00T + 73T^{2} \)
79 \( 1 + 13.3T + 79T^{2} \)
83 \( 1 - 8.30T + 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 - 9.62T + 97T^{2} \)
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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

−7.893845864500851963611175170817, −7.16747306297230020876064603585, −6.20200081268935148176485756096, −5.96073448898044910682063401756, −5.14159811007949523100525019551, −4.26074509102039710091784804739, −3.29690874335768691552042573860, −2.68684941155276941042052333811, −1.82076957964637101252894348918, −0.996345833357379752189259597592, 0.996345833357379752189259597592, 1.82076957964637101252894348918, 2.68684941155276941042052333811, 3.29690874335768691552042573860, 4.26074509102039710091784804739, 5.14159811007949523100525019551, 5.96073448898044910682063401756, 6.20200081268935148176485756096, 7.16747306297230020876064603585, 7.893845864500851963611175170817

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