Properties

Label 2-8016-1.1-c1-0-139
Degree $2$
Conductor $8016$
Sign $-1$
Analytic cond. $64.0080$
Root an. cond. $8.00050$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 0.621·5-s + 0.794·7-s + 9-s − 2.84·11-s + 2.63·13-s − 0.621·15-s − 3.05·17-s + 4.38·19-s + 0.794·21-s − 2.45·23-s − 4.61·25-s + 27-s − 3.48·29-s − 6.29·31-s − 2.84·33-s − 0.493·35-s + 5.11·37-s + 2.63·39-s + 9.70·41-s − 1.24·43-s − 0.621·45-s − 10.2·47-s − 6.36·49-s − 3.05·51-s + 4.00·53-s + 1.76·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.277·5-s + 0.300·7-s + 0.333·9-s − 0.856·11-s + 0.730·13-s − 0.160·15-s − 0.740·17-s + 1.00·19-s + 0.173·21-s − 0.512·23-s − 0.922·25-s + 0.192·27-s − 0.648·29-s − 1.13·31-s − 0.494·33-s − 0.0834·35-s + 0.840·37-s + 0.421·39-s + 1.51·41-s − 0.190·43-s − 0.0925·45-s − 1.49·47-s − 0.909·49-s − 0.427·51-s + 0.550·53-s + 0.238·55-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: $-1$
Analytic conductor: \(64.0080\)
Root analytic conductor: \(8.00050\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
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 \( 1 + 0.621T + 5T^{2} \)
7 \( 1 - 0.794T + 7T^{2} \)
11 \( 1 + 2.84T + 11T^{2} \)
13 \( 1 - 2.63T + 13T^{2} \)
17 \( 1 + 3.05T + 17T^{2} \)
19 \( 1 - 4.38T + 19T^{2} \)
23 \( 1 + 2.45T + 23T^{2} \)
29 \( 1 + 3.48T + 29T^{2} \)
31 \( 1 + 6.29T + 31T^{2} \)
37 \( 1 - 5.11T + 37T^{2} \)
41 \( 1 - 9.70T + 41T^{2} \)
43 \( 1 + 1.24T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 4.00T + 53T^{2} \)
59 \( 1 + 8.96T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 2.20T + 67T^{2} \)
71 \( 1 + 6.41T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 1.13T + 83T^{2} \)
89 \( 1 - 5.89T + 89T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−7.73339914616763218570816859162, −6.97437735210814490115484873689, −6.02025120541246705631930178234, −5.44556277171191546879427205542, −4.53421922015782995028984417140, −3.87130157451019421909707957120, −3.12113088083305001299666012938, −2.25843420312732843714400661492, −1.40651131417006142443635112465, 0, 1.40651131417006142443635112465, 2.25843420312732843714400661492, 3.12113088083305001299666012938, 3.87130157451019421909707957120, 4.53421922015782995028984417140, 5.44556277171191546879427205542, 6.02025120541246705631930178234, 6.97437735210814490115484873689, 7.73339914616763218570816859162

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