Properties

Label 2-6034-1.1-c1-0-45
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.743·3-s + 4-s − 1.77·5-s + 0.743·6-s + 7-s − 8-s − 2.44·9-s + 1.77·10-s + 2.78·11-s − 0.743·12-s + 5.15·13-s − 14-s + 1.31·15-s + 16-s + 0.832·17-s + 2.44·18-s + 4.69·19-s − 1.77·20-s − 0.743·21-s − 2.78·22-s + 5.67·23-s + 0.743·24-s − 1.86·25-s − 5.15·26-s + 4.04·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.429·3-s + 0.5·4-s − 0.791·5-s + 0.303·6-s + 0.377·7-s − 0.353·8-s − 0.815·9-s + 0.559·10-s + 0.839·11-s − 0.214·12-s + 1.42·13-s − 0.267·14-s + 0.339·15-s + 0.250·16-s + 0.202·17-s + 0.576·18-s + 1.07·19-s − 0.395·20-s − 0.162·21-s − 0.593·22-s + 1.18·23-s + 0.151·24-s − 0.373·25-s − 1.01·26-s + 0.779·27-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: \(0\)
Selberg data: \((2,\ 6034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.176007880\)
\(L(\frac12)\) \(\approx\) \(1.176007880\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.743T + 3T^{2} \)
5 \( 1 + 1.77T + 5T^{2} \)
11 \( 1 - 2.78T + 11T^{2} \)
13 \( 1 - 5.15T + 13T^{2} \)
17 \( 1 - 0.832T + 17T^{2} \)
19 \( 1 - 4.69T + 19T^{2} \)
23 \( 1 - 5.67T + 23T^{2} \)
29 \( 1 - 5.31T + 29T^{2} \)
31 \( 1 + 4.78T + 31T^{2} \)
37 \( 1 + 3.10T + 37T^{2} \)
41 \( 1 + 0.224T + 41T^{2} \)
43 \( 1 - 1.94T + 43T^{2} \)
47 \( 1 - 6.39T + 47T^{2} \)
53 \( 1 - 6.26T + 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 - 6.18T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 - 0.347T + 73T^{2} \)
79 \( 1 + 9.73T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + 6.60T + 89T^{2} \)
97 \( 1 + 2.27T + 97T^{2} \)
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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

−8.301517349725183640146765671873, −7.30249893611618845908961335531, −6.90811167800755133080984127187, −5.87612511445875088367834631474, −5.49532993197726685943334281375, −4.34439009983680991710301145842, −3.57021245670830623301881207550, −2.84480531918772781585912136792, −1.44078064140553870537851288241, −0.71742269629833668869462023090, 0.71742269629833668869462023090, 1.44078064140553870537851288241, 2.84480531918772781585912136792, 3.57021245670830623301881207550, 4.34439009983680991710301145842, 5.49532993197726685943334281375, 5.87612511445875088367834631474, 6.90811167800755133080984127187, 7.30249893611618845908961335531, 8.301517349725183640146765671873

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