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 − 2.99·5-s + 4.65·7-s + 9-s − 0.962·11-s + 0.976·13-s + 2.99·15-s + 2.56·17-s − 8.42·19-s − 4.65·21-s − 6.29·23-s + 3.96·25-s − 27-s − 3.04·29-s + 10.1·31-s + 0.962·33-s − 13.9·35-s + 10.7·37-s − 0.976·39-s + 0.413·41-s + 5.35·43-s − 2.99·45-s + 8.47·47-s + 14.7·49-s − 2.56·51-s − 13.1·53-s + 2.88·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.33·5-s + 1.76·7-s + 0.333·9-s − 0.290·11-s + 0.270·13-s + 0.773·15-s + 0.621·17-s − 1.93·19-s − 1.01·21-s − 1.31·23-s + 0.792·25-s − 0.192·27-s − 0.565·29-s + 1.82·31-s + 0.167·33-s − 2.35·35-s + 1.76·37-s − 0.156·39-s + 0.0645·41-s + 0.816·43-s − 0.446·45-s + 1.23·47-s + 2.10·49-s − 0.358·51-s − 1.80·53-s + 0.388·55-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

\(L(1)\) \(\approx\) \(1.275934514\)
\(L(\frac12)\) \(\approx\) \(1.275934514\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.99T + 5T^{2} \)
7 \( 1 - 4.65T + 7T^{2} \)
11 \( 1 + 0.962T + 11T^{2} \)
13 \( 1 - 0.976T + 13T^{2} \)
17 \( 1 - 2.56T + 17T^{2} \)
19 \( 1 + 8.42T + 19T^{2} \)
23 \( 1 + 6.29T + 23T^{2} \)
29 \( 1 + 3.04T + 29T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 - 0.413T + 41T^{2} \)
43 \( 1 - 5.35T + 43T^{2} \)
47 \( 1 - 8.47T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 - 0.101T + 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 - 1.47T + 79T^{2} \)
83 \( 1 - 4.56T + 83T^{2} \)
89 \( 1 - 8.20T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−7.79870571898195829099023802828, −7.50761835425765877417283529752, −6.27890683525787182144814564446, −5.86314478612341095333738802036, −4.67127455026622213887714494603, −4.47941073724319838155927757453, −3.85823146695728393365166738686, −2.58326646069481317137128628291, −1.64393695518311994111807661368, −0.59261146243357259774131289172, 0.59261146243357259774131289172, 1.64393695518311994111807661368, 2.58326646069481317137128628291, 3.85823146695728393365166738686, 4.47941073724319838155927757453, 4.67127455026622213887714494603, 5.86314478612341095333738802036, 6.27890683525787182144814564446, 7.50761835425765877417283529752, 7.79870571898195829099023802828

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