Properties

Label 2-8034-1.1-c1-0-3
Degree $2$
Conductor $8034$
Sign $1$
Analytic cond. $64.1518$
Root an. cond. $8.00948$
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  − 2-s − 3-s + 4-s − 3.43·5-s + 6-s − 0.875·7-s − 8-s + 9-s + 3.43·10-s − 4.12·11-s − 12-s − 13-s + 0.875·14-s + 3.43·15-s + 16-s + 2.91·17-s − 18-s + 6.27·19-s − 3.43·20-s + 0.875·21-s + 4.12·22-s − 2.22·23-s + 24-s + 6.81·25-s + 26-s − 27-s − 0.875·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.53·5-s + 0.408·6-s − 0.330·7-s − 0.353·8-s + 0.333·9-s + 1.08·10-s − 1.24·11-s − 0.288·12-s − 0.277·13-s + 0.234·14-s + 0.887·15-s + 0.250·16-s + 0.706·17-s − 0.235·18-s + 1.43·19-s − 0.768·20-s + 0.191·21-s + 0.880·22-s − 0.464·23-s + 0.204·24-s + 1.36·25-s + 0.196·26-s − 0.192·27-s − 0.165·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
Sign: $1$
Analytic conductor: \(64.1518\)
Root analytic conductor: \(8.00948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2785838900\)
\(L(\frac12)\) \(\approx\) \(0.2785838900\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
13 \( 1 + T \)
103 \( 1 - T \)
good5 \( 1 + 3.43T + 5T^{2} \)
7 \( 1 + 0.875T + 7T^{2} \)
11 \( 1 + 4.12T + 11T^{2} \)
17 \( 1 - 2.91T + 17T^{2} \)
19 \( 1 - 6.27T + 19T^{2} \)
23 \( 1 + 2.22T + 23T^{2} \)
29 \( 1 + 3.54T + 29T^{2} \)
31 \( 1 - 4.78T + 31T^{2} \)
37 \( 1 - 1.66T + 37T^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 + 7.01T + 43T^{2} \)
47 \( 1 - 7.14T + 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 + 7.72T + 59T^{2} \)
61 \( 1 + 2.25T + 61T^{2} \)
67 \( 1 + 5.69T + 67T^{2} \)
71 \( 1 + 0.740T + 71T^{2} \)
73 \( 1 - 5.84T + 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 - 3.84T + 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−7.79723907784136109263029735605, −7.41926217876655982354087006405, −6.66775494634936895240423626894, −5.74567810269351740502956900704, −5.09723994294327699191927821565, −4.31316725681119835651136358612, −3.32763155216261600561771709714, −2.85548544470934900589785544821, −1.42141121113758165444200392351, −0.31285552070410861934162807343, 0.31285552070410861934162807343, 1.42141121113758165444200392351, 2.85548544470934900589785544821, 3.32763155216261600561771709714, 4.31316725681119835651136358612, 5.09723994294327699191927821565, 5.74567810269351740502956900704, 6.66775494634936895240423626894, 7.41926217876655982354087006405, 7.79723907784136109263029735605

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