Properties

Label 2-8034-1.1-c1-0-134
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 + 4.22·5-s + 6-s − 0.979·7-s + 8-s + 9-s + 4.22·10-s + 1.72·11-s + 12-s + 13-s − 0.979·14-s + 4.22·15-s + 16-s − 2.27·17-s + 18-s − 1.97·19-s + 4.22·20-s − 0.979·21-s + 1.72·22-s + 1.31·23-s + 24-s + 12.8·25-s + 26-s + 27-s − 0.979·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.88·5-s + 0.408·6-s − 0.370·7-s + 0.353·8-s + 0.333·9-s + 1.33·10-s + 0.519·11-s + 0.288·12-s + 0.277·13-s − 0.261·14-s + 1.09·15-s + 0.250·16-s − 0.550·17-s + 0.235·18-s − 0.454·19-s + 0.944·20-s − 0.213·21-s + 0.367·22-s + 0.273·23-s + 0.204·24-s + 2.56·25-s + 0.196·26-s + 0.192·27-s − 0.185·28-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

\(L(1)\) \(\approx\) \(6.446936809\)
\(L(\frac12)\) \(\approx\) \(6.446936809\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.22T + 5T^{2} \)
7 \( 1 + 0.979T + 7T^{2} \)
11 \( 1 - 1.72T + 11T^{2} \)
17 \( 1 + 2.27T + 17T^{2} \)
19 \( 1 + 1.97T + 19T^{2} \)
23 \( 1 - 1.31T + 23T^{2} \)
29 \( 1 - 9.39T + 29T^{2} \)
31 \( 1 - 1.97T + 31T^{2} \)
37 \( 1 - 11.5T + 37T^{2} \)
41 \( 1 - 4.25T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + 4.84T + 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 - 1.57T + 59T^{2} \)
61 \( 1 + 8.99T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 5.91T + 73T^{2} \)
79 \( 1 + 8.55T + 79T^{2} \)
83 \( 1 - 7.40T + 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−7.80642319321901049220640581783, −6.73609056640165442860530348004, −6.30170128011914261096004430871, −6.03548477203638459926711534565, −4.81475876771048455598505409275, −4.55933024488524010087663264420, −3.24864123765960477878883715551, −2.76848882622406580841180289419, −1.93559769420035452853972063146, −1.21549216585654728629726439143, 1.21549216585654728629726439143, 1.93559769420035452853972063146, 2.76848882622406580841180289419, 3.24864123765960477878883715551, 4.55933024488524010087663264420, 4.81475876771048455598505409275, 6.03548477203638459926711534565, 6.30170128011914261096004430871, 6.73609056640165442860530348004, 7.80642319321901049220640581783

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