Properties

Label 2-6014-1.1-c1-0-50
Degree $2$
Conductor $6014$
Sign $1$
Analytic cond. $48.0220$
Root an. cond. $6.92979$
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 − 0.0623·3-s + 4-s + 1.97·5-s + 0.0623·6-s − 0.185·7-s − 8-s − 2.99·9-s − 1.97·10-s − 1.12·11-s − 0.0623·12-s − 4.71·13-s + 0.185·14-s − 0.123·15-s + 16-s + 5.95·17-s + 2.99·18-s − 5.93·19-s + 1.97·20-s + 0.0115·21-s + 1.12·22-s + 5.58·23-s + 0.0623·24-s − 1.10·25-s + 4.71·26-s + 0.373·27-s − 0.185·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.0359·3-s + 0.5·4-s + 0.883·5-s + 0.0254·6-s − 0.0701·7-s − 0.353·8-s − 0.998·9-s − 0.624·10-s − 0.338·11-s − 0.0179·12-s − 1.30·13-s + 0.0496·14-s − 0.0317·15-s + 0.250·16-s + 1.44·17-s + 0.706·18-s − 1.36·19-s + 0.441·20-s + 0.00252·21-s + 0.239·22-s + 1.16·23-s + 0.0127·24-s − 0.220·25-s + 0.924·26-s + 0.0719·27-s − 0.0350·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6014\)    =    \(2 \cdot 31 \cdot 97\)
Sign: $1$
Analytic conductor: \(48.0220\)
Root analytic conductor: \(6.92979\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6014,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.173366640\)
\(L(\frac12)\) \(\approx\) \(1.173366640\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
31 \( 1 + T \)
97 \( 1 - T \)
good3 \( 1 + 0.0623T + 3T^{2} \)
5 \( 1 - 1.97T + 5T^{2} \)
7 \( 1 + 0.185T + 7T^{2} \)
11 \( 1 + 1.12T + 11T^{2} \)
13 \( 1 + 4.71T + 13T^{2} \)
17 \( 1 - 5.95T + 17T^{2} \)
19 \( 1 + 5.93T + 19T^{2} \)
23 \( 1 - 5.58T + 23T^{2} \)
29 \( 1 - 3.59T + 29T^{2} \)
37 \( 1 - 5.72T + 37T^{2} \)
41 \( 1 - 6.21T + 41T^{2} \)
43 \( 1 + 5.51T + 43T^{2} \)
47 \( 1 + 9.14T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + 4.49T + 59T^{2} \)
61 \( 1 + 9.23T + 61T^{2} \)
67 \( 1 - 15.9T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 - 1.46T + 73T^{2} \)
79 \( 1 + 17.4T + 79T^{2} \)
83 \( 1 + 12.4T + 83T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
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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

−8.133719882751753170803868332702, −7.50980058084933410694223463119, −6.65327635670223187092200063158, −6.01017971446729134948164271271, −5.36127362048983821337568176089, −4.66804222408589103228299485902, −3.27724904705104646981626821645, −2.62339628095552761826606721735, −1.89350536153172085554481984881, −0.61178450423065832554913318428, 0.61178450423065832554913318428, 1.89350536153172085554481984881, 2.62339628095552761826606721735, 3.27724904705104646981626821645, 4.66804222408589103228299485902, 5.36127362048983821337568176089, 6.01017971446729134948164271271, 6.65327635670223187092200063158, 7.50980058084933410694223463119, 8.133719882751753170803868332702

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