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 − 0.858·5-s − 2.33·7-s + 9-s − 0.664·11-s − 2.80·13-s + 0.858·15-s − 4.73·17-s + 7.58·19-s + 2.33·21-s − 6.58·23-s − 4.26·25-s − 27-s − 4.75·29-s + 2.96·31-s + 0.664·33-s + 2.00·35-s + 4.12·37-s + 2.80·39-s − 7.76·41-s − 4.13·43-s − 0.858·45-s + 1.39·47-s − 1.56·49-s + 4.73·51-s − 1.74·53-s + 0.569·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.383·5-s − 0.881·7-s + 0.333·9-s − 0.200·11-s − 0.777·13-s + 0.221·15-s − 1.14·17-s + 1.74·19-s + 0.508·21-s − 1.37·23-s − 0.852·25-s − 0.192·27-s − 0.883·29-s + 0.531·31-s + 0.115·33-s + 0.338·35-s + 0.678·37-s + 0.449·39-s − 1.21·41-s − 0.631·43-s − 0.127·45-s + 0.202·47-s − 0.222·49-s + 0.663·51-s − 0.239·53-s + 0.0768·55-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

\(L(1)\) \(\approx\) \(0.5031433659\)
\(L(\frac12)\) \(\approx\) \(0.5031433659\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.858T + 5T^{2} \)
7 \( 1 + 2.33T + 7T^{2} \)
11 \( 1 + 0.664T + 11T^{2} \)
13 \( 1 + 2.80T + 13T^{2} \)
17 \( 1 + 4.73T + 17T^{2} \)
19 \( 1 - 7.58T + 19T^{2} \)
23 \( 1 + 6.58T + 23T^{2} \)
29 \( 1 + 4.75T + 29T^{2} \)
31 \( 1 - 2.96T + 31T^{2} \)
37 \( 1 - 4.12T + 37T^{2} \)
41 \( 1 + 7.76T + 41T^{2} \)
43 \( 1 + 4.13T + 43T^{2} \)
47 \( 1 - 1.39T + 47T^{2} \)
53 \( 1 + 1.74T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + 0.762T + 61T^{2} \)
67 \( 1 + 0.00130T + 67T^{2} \)
71 \( 1 - 2.99T + 71T^{2} \)
73 \( 1 + 16.6T + 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 + 17.4T + 83T^{2} \)
89 \( 1 + 4.94T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−7.67672344457698274890172840307, −7.14139399771106204927139822866, −6.45479929458895314679252856919, −5.76811356640078828306388161017, −5.10682206317893254087349760717, −4.27994197801824096183830694600, −3.58400827666068625535196610984, −2.72430836331412888427472473847, −1.74740656008343010575332156701, −0.34829004676635298897594167364, 0.34829004676635298897594167364, 1.74740656008343010575332156701, 2.72430836331412888427472473847, 3.58400827666068625535196610984, 4.27994197801824096183830694600, 5.10682206317893254087349760717, 5.76811356640078828306388161017, 6.45479929458895314679252856919, 7.14139399771106204927139822866, 7.67672344457698274890172840307

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