Properties

Degree $2$
Conductor $8016$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 1.82·5-s − 2.26·7-s + 9-s + 0.576·11-s − 1.28·13-s + 1.82·15-s − 1.88·17-s − 5.66·19-s − 2.26·21-s + 2.15·23-s − 1.66·25-s + 27-s + 6.70·29-s − 2.07·31-s + 0.576·33-s − 4.14·35-s − 5.63·37-s − 1.28·39-s + 8.23·41-s − 2.54·43-s + 1.82·45-s − 9.37·47-s − 1.84·49-s − 1.88·51-s + 6.17·53-s + 1.05·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.816·5-s − 0.857·7-s + 0.333·9-s + 0.173·11-s − 0.355·13-s + 0.471·15-s − 0.456·17-s − 1.29·19-s − 0.495·21-s + 0.449·23-s − 0.332·25-s + 0.192·27-s + 1.24·29-s − 0.372·31-s + 0.100·33-s − 0.700·35-s − 0.926·37-s − 0.205·39-s + 1.28·41-s − 0.388·43-s + 0.272·45-s − 1.36·47-s − 0.264·49-s − 0.263·51-s + 0.848·53-s + 0.141·55-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: \(1\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.82T + 5T^{2} \)
7 \( 1 + 2.26T + 7T^{2} \)
11 \( 1 - 0.576T + 11T^{2} \)
13 \( 1 + 1.28T + 13T^{2} \)
17 \( 1 + 1.88T + 17T^{2} \)
19 \( 1 + 5.66T + 19T^{2} \)
23 \( 1 - 2.15T + 23T^{2} \)
29 \( 1 - 6.70T + 29T^{2} \)
31 \( 1 + 2.07T + 31T^{2} \)
37 \( 1 + 5.63T + 37T^{2} \)
41 \( 1 - 8.23T + 41T^{2} \)
43 \( 1 + 2.54T + 43T^{2} \)
47 \( 1 + 9.37T + 47T^{2} \)
53 \( 1 - 6.17T + 53T^{2} \)
59 \( 1 - 3.25T + 59T^{2} \)
61 \( 1 - 1.95T + 61T^{2} \)
67 \( 1 + 8.37T + 67T^{2} \)
71 \( 1 - 2.61T + 71T^{2} \)
73 \( 1 + 5.82T + 73T^{2} \)
79 \( 1 - 1.10T + 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 + 3.01T + 89T^{2} \)
97 \( 1 + 5.58T + 97T^{2} \)
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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

−7.39824793347898235416004593275, −6.68460279126830487651733874171, −6.29493150513934428120131310272, −5.46006811073047706869999706549, −4.58104496370043598564475955285, −3.86923035092990271475285611739, −2.93727634776662109083862804418, −2.34503180134762693604375030588, −1.46277656236414483271641951737, 0, 1.46277656236414483271641951737, 2.34503180134762693604375030588, 2.93727634776662109083862804418, 3.86923035092990271475285611739, 4.58104496370043598564475955285, 5.46006811073047706869999706549, 6.29493150513934428120131310272, 6.68460279126830487651733874171, 7.39824793347898235416004593275

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