Properties

Label 2-6034-1.1-c1-0-147
Degree $2$
Conductor $6034$
Sign $-1$
Analytic cond. $48.1817$
Root an. cond. $6.94130$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.286·3-s + 4-s + 3.05·5-s + 0.286·6-s − 7-s − 8-s − 2.91·9-s − 3.05·10-s − 1.07·11-s − 0.286·12-s + 0.416·13-s + 14-s − 0.876·15-s + 16-s + 3.47·17-s + 2.91·18-s − 8.09·19-s + 3.05·20-s + 0.286·21-s + 1.07·22-s + 1.29·23-s + 0.286·24-s + 4.35·25-s − 0.416·26-s + 1.69·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.165·3-s + 0.5·4-s + 1.36·5-s + 0.116·6-s − 0.377·7-s − 0.353·8-s − 0.972·9-s − 0.967·10-s − 0.324·11-s − 0.0826·12-s + 0.115·13-s + 0.267·14-s − 0.226·15-s + 0.250·16-s + 0.841·17-s + 0.687·18-s − 1.85·19-s + 0.683·20-s + 0.0625·21-s + 0.229·22-s + 0.270·23-s + 0.0584·24-s + 0.871·25-s − 0.0816·26-s + 0.326·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6034\)    =    \(2 \cdot 7 \cdot 431\)
Sign: $-1$
Analytic conductor: \(48.1817\)
Root analytic conductor: \(6.94130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6034,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 + T \)
431 \( 1 + T \)
good3 \( 1 + 0.286T + 3T^{2} \)
5 \( 1 - 3.05T + 5T^{2} \)
11 \( 1 + 1.07T + 11T^{2} \)
13 \( 1 - 0.416T + 13T^{2} \)
17 \( 1 - 3.47T + 17T^{2} \)
19 \( 1 + 8.09T + 19T^{2} \)
23 \( 1 - 1.29T + 23T^{2} \)
29 \( 1 - 3.37T + 29T^{2} \)
31 \( 1 - 0.387T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 - 9.90T + 41T^{2} \)
43 \( 1 + 1.60T + 43T^{2} \)
47 \( 1 - 0.853T + 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 + 3.69T + 59T^{2} \)
61 \( 1 - 2.58T + 61T^{2} \)
67 \( 1 + 7.58T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + 1.99T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + 5.26T + 83T^{2} \)
89 \( 1 + 9.10T + 89T^{2} \)
97 \( 1 + 6.49T + 97T^{2} \)
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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

−7.84892301694262409160855319977, −6.88247562623931528963041029726, −6.30189932527749548601253310483, −5.73017074570907231567412335892, −5.18349943558813232006314453165, −3.95433321437759480976405005952, −2.80853250418792692173131630273, −2.33628262649904247133992061764, −1.29619976795849948340652520042, 0, 1.29619976795849948340652520042, 2.33628262649904247133992061764, 2.80853250418792692173131630273, 3.95433321437759480976405005952, 5.18349943558813232006314453165, 5.73017074570907231567412335892, 6.30189932527749548601253310483, 6.88247562623931528963041029726, 7.84892301694262409160855319977

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