Properties

Label 2-8016-1.1-c1-0-66
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 + 1.52·5-s + 1.05·7-s + 9-s − 0.162·11-s + 3.63·13-s − 1.52·15-s + 4.89·17-s − 1.02·19-s − 1.05·21-s + 7.59·23-s − 2.65·25-s − 27-s + 9.59·29-s − 2.68·31-s + 0.162·33-s + 1.61·35-s + 4.12·37-s − 3.63·39-s − 1.08·41-s − 2.77·43-s + 1.52·45-s + 9.50·47-s − 5.89·49-s − 4.89·51-s + 4.80·53-s − 0.247·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.684·5-s + 0.397·7-s + 0.333·9-s − 0.0488·11-s + 1.00·13-s − 0.395·15-s + 1.18·17-s − 0.235·19-s − 0.229·21-s + 1.58·23-s − 0.531·25-s − 0.192·27-s + 1.78·29-s − 0.481·31-s + 0.0282·33-s + 0.272·35-s + 0.677·37-s − 0.582·39-s − 0.168·41-s − 0.423·43-s + 0.228·45-s + 1.38·47-s − 0.841·49-s − 0.685·51-s + 0.660·53-s − 0.0334·55-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

\(L(1)\) \(\approx\) \(2.481460783\)
\(L(\frac12)\) \(\approx\) \(2.481460783\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.52T + 5T^{2} \)
7 \( 1 - 1.05T + 7T^{2} \)
11 \( 1 + 0.162T + 11T^{2} \)
13 \( 1 - 3.63T + 13T^{2} \)
17 \( 1 - 4.89T + 17T^{2} \)
19 \( 1 + 1.02T + 19T^{2} \)
23 \( 1 - 7.59T + 23T^{2} \)
29 \( 1 - 9.59T + 29T^{2} \)
31 \( 1 + 2.68T + 31T^{2} \)
37 \( 1 - 4.12T + 37T^{2} \)
41 \( 1 + 1.08T + 41T^{2} \)
43 \( 1 + 2.77T + 43T^{2} \)
47 \( 1 - 9.50T + 47T^{2} \)
53 \( 1 - 4.80T + 53T^{2} \)
59 \( 1 + 8.22T + 59T^{2} \)
61 \( 1 - 0.510T + 61T^{2} \)
67 \( 1 - 0.317T + 67T^{2} \)
71 \( 1 + 5.83T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 + 5.59T + 79T^{2} \)
83 \( 1 + 4.98T + 83T^{2} \)
89 \( 1 + 0.199T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−7.82020864508692752254477537728, −7.04519073346196807281921742843, −6.32420955600555322712544818983, −5.77323534536945328591810517650, −5.15037981603515264101705075661, −4.44518726921924039631344892778, −3.50024853715303347669658885993, −2.66671891052833934729302760557, −1.53443944429900098128924125306, −0.892792182216449293807473103113, 0.892792182216449293807473103113, 1.53443944429900098128924125306, 2.66671891052833934729302760557, 3.50024853715303347669658885993, 4.44518726921924039631344892778, 5.15037981603515264101705075661, 5.77323534536945328591810517650, 6.32420955600555322712544818983, 7.04519073346196807281921742843, 7.82020864508692752254477537728

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