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 + 4.01·5-s + 4.41·7-s + 9-s − 3.91·11-s + 1.33·13-s + 4.01·15-s + 5.36·17-s − 4.32·19-s + 4.41·21-s − 2.96·23-s + 11.1·25-s + 27-s + 0.641·29-s + 4.50·31-s − 3.91·33-s + 17.7·35-s − 3.58·37-s + 1.33·39-s − 8.78·41-s − 6.83·43-s + 4.01·45-s + 11.1·47-s + 12.4·49-s + 5.36·51-s − 1.90·53-s − 15.7·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.79·5-s + 1.66·7-s + 0.333·9-s − 1.18·11-s + 0.369·13-s + 1.03·15-s + 1.30·17-s − 0.991·19-s + 0.963·21-s − 0.617·23-s + 2.22·25-s + 0.192·27-s + 0.119·29-s + 0.808·31-s − 0.682·33-s + 2.99·35-s − 0.589·37-s + 0.213·39-s − 1.37·41-s − 1.04·43-s + 0.598·45-s + 1.62·47-s + 1.78·49-s + 0.751·51-s − 0.262·53-s − 2.12·55-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

\(L(1)\) \(\approx\) \(4.738504628\)
\(L(\frac12)\) \(\approx\) \(4.738504628\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.01T + 5T^{2} \)
7 \( 1 - 4.41T + 7T^{2} \)
11 \( 1 + 3.91T + 11T^{2} \)
13 \( 1 - 1.33T + 13T^{2} \)
17 \( 1 - 5.36T + 17T^{2} \)
19 \( 1 + 4.32T + 19T^{2} \)
23 \( 1 + 2.96T + 23T^{2} \)
29 \( 1 - 0.641T + 29T^{2} \)
31 \( 1 - 4.50T + 31T^{2} \)
37 \( 1 + 3.58T + 37T^{2} \)
41 \( 1 + 8.78T + 41T^{2} \)
43 \( 1 + 6.83T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 1.90T + 53T^{2} \)
59 \( 1 - 6.60T + 59T^{2} \)
61 \( 1 - 0.00823T + 61T^{2} \)
67 \( 1 + 4.03T + 67T^{2} \)
71 \( 1 - 2.34T + 71T^{2} \)
73 \( 1 - 4.99T + 73T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−8.029875070486760102925512419294, −7.25371316501171039711447678504, −6.34259896452402327644451659695, −5.59966391539160028049371140028, −5.15608193375901557893672805528, −4.50092081905103943410209323473, −3.33393136535703631515569519456, −2.35873449901541051266872847303, −1.93114431428867355485984028399, −1.14934157163763247527317651852, 1.14934157163763247527317651852, 1.93114431428867355485984028399, 2.35873449901541051266872847303, 3.33393136535703631515569519456, 4.50092081905103943410209323473, 5.15608193375901557893672805528, 5.59966391539160028049371140028, 6.34259896452402327644451659695, 7.25371316501171039711447678504, 8.029875070486760102925512419294

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