Properties

Label 2-8016-1.1-c1-0-36
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 + 3.42·5-s − 3.44·7-s + 9-s + 3.41·11-s − 2.00·13-s − 3.42·15-s − 3.07·17-s + 0.565·19-s + 3.44·21-s − 7.40·23-s + 6.76·25-s − 27-s + 9.73·29-s + 2.62·31-s − 3.41·33-s − 11.8·35-s − 0.527·37-s + 2.00·39-s + 3.19·41-s − 7.98·43-s + 3.42·45-s + 6.96·47-s + 4.84·49-s + 3.07·51-s + 2.40·53-s + 11.7·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.53·5-s − 1.30·7-s + 0.333·9-s + 1.02·11-s − 0.556·13-s − 0.885·15-s − 0.746·17-s + 0.129·19-s + 0.751·21-s − 1.54·23-s + 1.35·25-s − 0.192·27-s + 1.80·29-s + 0.471·31-s − 0.593·33-s − 1.99·35-s − 0.0866·37-s + 0.321·39-s + 0.499·41-s − 1.21·43-s + 0.511·45-s + 1.01·47-s + 0.692·49-s + 0.430·51-s + 0.330·53-s + 1.57·55-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: $\chi_{8016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.798118492\)
\(L(\frac12)\) \(\approx\) \(1.798118492\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.42T + 5T^{2} \)
7 \( 1 + 3.44T + 7T^{2} \)
11 \( 1 - 3.41T + 11T^{2} \)
13 \( 1 + 2.00T + 13T^{2} \)
17 \( 1 + 3.07T + 17T^{2} \)
19 \( 1 - 0.565T + 19T^{2} \)
23 \( 1 + 7.40T + 23T^{2} \)
29 \( 1 - 9.73T + 29T^{2} \)
31 \( 1 - 2.62T + 31T^{2} \)
37 \( 1 + 0.527T + 37T^{2} \)
41 \( 1 - 3.19T + 41T^{2} \)
43 \( 1 + 7.98T + 43T^{2} \)
47 \( 1 - 6.96T + 47T^{2} \)
53 \( 1 - 2.40T + 53T^{2} \)
59 \( 1 + 6.23T + 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 + 0.623T + 67T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 - 8.13T + 73T^{2} \)
79 \( 1 - 4.25T + 79T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 - 3.69T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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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

−7.71463282298998300961872071722, −6.67285767269346485954178256443, −6.37811935119215727008963146736, −6.08136954442910256911510589695, −5.12318188689092817603672474054, −4.41645466831236084010772720096, −3.47285908288475013179070647817, −2.53454766841509965318641306742, −1.81704421664154926594113129424, −0.67674519476775632653716304034, 0.67674519476775632653716304034, 1.81704421664154926594113129424, 2.53454766841509965318641306742, 3.47285908288475013179070647817, 4.41645466831236084010772720096, 5.12318188689092817603672474054, 6.08136954442910256911510589695, 6.37811935119215727008963146736, 6.67285767269346485954178256443, 7.71463282298998300961872071722

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