Properties

Label 2-8016-1.1-c1-0-48
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 0.284·5-s + 3.90·7-s + 9-s − 5.93·11-s + 5.62·13-s − 0.284·15-s − 0.211·17-s + 1.66·19-s − 3.90·21-s + 2.16·23-s − 4.91·25-s − 27-s + 4.18·29-s + 2.41·31-s + 5.93·33-s + 1.10·35-s + 0.0651·37-s − 5.62·39-s + 6.23·41-s − 2.49·43-s + 0.284·45-s − 5.97·47-s + 8.26·49-s + 0.211·51-s + 4.43·53-s − 1.68·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.127·5-s + 1.47·7-s + 0.333·9-s − 1.79·11-s + 1.55·13-s − 0.0733·15-s − 0.0513·17-s + 0.381·19-s − 0.852·21-s + 0.451·23-s − 0.983·25-s − 0.192·27-s + 0.777·29-s + 0.433·31-s + 1.03·33-s + 0.187·35-s + 0.0107·37-s − 0.900·39-s + 0.973·41-s − 0.379·43-s + 0.0423·45-s − 0.871·47-s + 1.18·49-s + 0.0296·51-s + 0.609·53-s − 0.227·55-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: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.047222902\)
\(L(\frac12)\) \(\approx\) \(2.047222902\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.284T + 5T^{2} \)
7 \( 1 - 3.90T + 7T^{2} \)
11 \( 1 + 5.93T + 11T^{2} \)
13 \( 1 - 5.62T + 13T^{2} \)
17 \( 1 + 0.211T + 17T^{2} \)
19 \( 1 - 1.66T + 19T^{2} \)
23 \( 1 - 2.16T + 23T^{2} \)
29 \( 1 - 4.18T + 29T^{2} \)
31 \( 1 - 2.41T + 31T^{2} \)
37 \( 1 - 0.0651T + 37T^{2} \)
41 \( 1 - 6.23T + 41T^{2} \)
43 \( 1 + 2.49T + 43T^{2} \)
47 \( 1 + 5.97T + 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 - 3.03T + 59T^{2} \)
61 \( 1 + 5.18T + 61T^{2} \)
67 \( 1 - 5.08T + 67T^{2} \)
71 \( 1 + 6.06T + 71T^{2} \)
73 \( 1 - 8.31T + 73T^{2} \)
79 \( 1 - 2.87T + 79T^{2} \)
83 \( 1 - 5.50T + 83T^{2} \)
89 \( 1 - 0.660T + 89T^{2} \)
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

−8.032907854615204341709322122197, −7.25123076511011952869125509945, −6.28769241401949815932927609794, −5.66191425830925772843781761454, −5.07167210503813976997706473216, −4.53657291950723545230548380293, −3.57739137548422459244697102600, −2.55023445099100822854258413557, −1.66952229967608476220811418001, −0.76361153578716861587058377981, 0.76361153578716861587058377981, 1.66952229967608476220811418001, 2.55023445099100822854258413557, 3.57739137548422459244697102600, 4.53657291950723545230548380293, 5.07167210503813976997706473216, 5.66191425830925772843781761454, 6.28769241401949815932927609794, 7.25123076511011952869125509945, 8.032907854615204341709322122197

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