Properties

Label 2-8034-1.1-c1-0-150
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 + 4.01·5-s + 6-s + 2.23·7-s + 8-s + 9-s + 4.01·10-s − 5.19·11-s + 12-s + 13-s + 2.23·14-s + 4.01·15-s + 16-s + 7.80·17-s + 18-s + 1.23·19-s + 4.01·20-s + 2.23·21-s − 5.19·22-s − 2.23·23-s + 24-s + 11.1·25-s + 26-s + 27-s + 2.23·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.79·5-s + 0.408·6-s + 0.845·7-s + 0.353·8-s + 0.333·9-s + 1.26·10-s − 1.56·11-s + 0.288·12-s + 0.277·13-s + 0.597·14-s + 1.03·15-s + 0.250·16-s + 1.89·17-s + 0.235·18-s + 0.283·19-s + 0.897·20-s + 0.488·21-s − 1.10·22-s − 0.465·23-s + 0.204·24-s + 2.22·25-s + 0.196·26-s + 0.192·27-s + 0.422·28-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

\(L(1)\) \(\approx\) \(6.787940848\)
\(L(\frac12)\) \(\approx\) \(6.787940848\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.01T + 5T^{2} \)
7 \( 1 - 2.23T + 7T^{2} \)
11 \( 1 + 5.19T + 11T^{2} \)
17 \( 1 - 7.80T + 17T^{2} \)
19 \( 1 - 1.23T + 19T^{2} \)
23 \( 1 + 2.23T + 23T^{2} \)
29 \( 1 + 0.116T + 29T^{2} \)
31 \( 1 + 1.23T + 31T^{2} \)
37 \( 1 + 4.93T + 37T^{2} \)
41 \( 1 - 3.24T + 41T^{2} \)
43 \( 1 - 9.31T + 43T^{2} \)
47 \( 1 - 1.50T + 47T^{2} \)
53 \( 1 + 2.85T + 53T^{2} \)
59 \( 1 + 9.33T + 59T^{2} \)
61 \( 1 + 1.08T + 61T^{2} \)
67 \( 1 + 5.69T + 67T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 - 0.0792T + 79T^{2} \)
83 \( 1 + 9.59T + 83T^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 + 3.88T + 97T^{2} \)
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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

−7.66377171812084768119867015914, −7.31883159776434256739733531950, −6.02440571356840801172787726508, −5.75012302155413169992263958793, −5.14595752475856931079527689683, −4.48713141250674054784751647812, −3.23677727073889117318322690724, −2.73784315737658300870766037836, −1.90140802492242869210670232971, −1.27817084898797039268920571187, 1.27817084898797039268920571187, 1.90140802492242869210670232971, 2.73784315737658300870766037836, 3.23677727073889117318322690724, 4.48713141250674054784751647812, 5.14595752475856931079527689683, 5.75012302155413169992263958793, 6.02440571356840801172787726508, 7.31883159776434256739733531950, 7.66377171812084768119867015914

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