Properties

Degree $2$
Conductor $8016$
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 − 1.98·5-s + 0.0909·7-s + 9-s + 2.35·11-s + 0.834·13-s + 1.98·15-s + 7.08·17-s + 1.66·19-s − 0.0909·21-s + 2.49·23-s − 1.05·25-s − 27-s − 3.96·29-s + 3.52·31-s − 2.35·33-s − 0.180·35-s − 1.64·37-s − 0.834·39-s + 0.142·41-s + 5.36·43-s − 1.98·45-s + 6.35·47-s − 6.99·49-s − 7.08·51-s + 13.4·53-s − 4.67·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.888·5-s + 0.0343·7-s + 0.333·9-s + 0.710·11-s + 0.231·13-s + 0.512·15-s + 1.71·17-s + 0.382·19-s − 0.0198·21-s + 0.521·23-s − 0.211·25-s − 0.192·27-s − 0.736·29-s + 0.633·31-s − 0.410·33-s − 0.0305·35-s − 0.270·37-s − 0.133·39-s + 0.0223·41-s + 0.817·43-s − 0.296·45-s + 0.926·47-s − 0.998·49-s − 0.992·51-s + 1.84·53-s − 0.630·55-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

Degree: \(2\)
Conductor: \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{8016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.527350887\)
\(L(\frac12)\) \(\approx\) \(1.527350887\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
167 \( 1 - T \)
good5 \( 1 + 1.98T + 5T^{2} \)
7 \( 1 - 0.0909T + 7T^{2} \)
11 \( 1 - 2.35T + 11T^{2} \)
13 \( 1 - 0.834T + 13T^{2} \)
17 \( 1 - 7.08T + 17T^{2} \)
19 \( 1 - 1.66T + 19T^{2} \)
23 \( 1 - 2.49T + 23T^{2} \)
29 \( 1 + 3.96T + 29T^{2} \)
31 \( 1 - 3.52T + 31T^{2} \)
37 \( 1 + 1.64T + 37T^{2} \)
41 \( 1 - 0.142T + 41T^{2} \)
43 \( 1 - 5.36T + 43T^{2} \)
47 \( 1 - 6.35T + 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + 4.39T + 59T^{2} \)
61 \( 1 - 0.0766T + 61T^{2} \)
67 \( 1 + 4.51T + 67T^{2} \)
71 \( 1 + 3.17T + 71T^{2} \)
73 \( 1 - 9.30T + 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 + 0.0141T + 83T^{2} \)
89 \( 1 + 7.96T + 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.74050077302514111394758986334, −7.22259136380577446155683938272, −6.47335733623251119540311938582, −5.66669365948957332435790541503, −5.16448968271830723901626456385, −4.11713171488932235203769215524, −3.72646209815349392702558920445, −2.83087890976948697750348406517, −1.47833025938940282960356339080, −0.68748175377681242193768948320, 0.68748175377681242193768948320, 1.47833025938940282960356339080, 2.83087890976948697750348406517, 3.72646209815349392702558920445, 4.11713171488932235203769215524, 5.16448968271830723901626456385, 5.66669365948957332435790541503, 6.47335733623251119540311938582, 7.22259136380577446155683938272, 7.74050077302514111394758986334

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