Properties

Label 2-6014-1.1-c1-0-138
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.51·3-s + 4-s + 3.35·5-s + 2.51·6-s + 0.494·7-s − 8-s + 3.32·9-s − 3.35·10-s + 0.0984·11-s − 2.51·12-s − 3.30·13-s − 0.494·14-s − 8.43·15-s + 16-s − 7.85·17-s − 3.32·18-s − 3.01·19-s + 3.35·20-s − 1.24·21-s − 0.0984·22-s + 7.95·23-s + 2.51·24-s + 6.25·25-s + 3.30·26-s − 0.811·27-s + 0.494·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.45·3-s + 0.5·4-s + 1.50·5-s + 1.02·6-s + 0.187·7-s − 0.353·8-s + 1.10·9-s − 1.06·10-s + 0.0296·11-s − 0.725·12-s − 0.917·13-s − 0.132·14-s − 2.17·15-s + 0.250·16-s − 1.90·17-s − 0.783·18-s − 0.690·19-s + 0.750·20-s − 0.271·21-s − 0.0209·22-s + 1.65·23-s + 0.513·24-s + 1.25·25-s + 0.649·26-s − 0.156·27-s + 0.0935·28-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: \(1\)
Selberg data: \((2,\ 6014,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
31 \( 1 - T \)
97 \( 1 - T \)
good3 \( 1 + 2.51T + 3T^{2} \)
5 \( 1 - 3.35T + 5T^{2} \)
7 \( 1 - 0.494T + 7T^{2} \)
11 \( 1 - 0.0984T + 11T^{2} \)
13 \( 1 + 3.30T + 13T^{2} \)
17 \( 1 + 7.85T + 17T^{2} \)
19 \( 1 + 3.01T + 19T^{2} \)
23 \( 1 - 7.95T + 23T^{2} \)
29 \( 1 - 1.21T + 29T^{2} \)
37 \( 1 - 7.83T + 37T^{2} \)
41 \( 1 - 3.22T + 41T^{2} \)
43 \( 1 - 2.99T + 43T^{2} \)
47 \( 1 + 8.11T + 47T^{2} \)
53 \( 1 - 6.70T + 53T^{2} \)
59 \( 1 + 8.38T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 14.9T + 67T^{2} \)
71 \( 1 - 3.75T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 - 6.85T + 79T^{2} \)
83 \( 1 - 5.78T + 83T^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
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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

−7.56055034673341401976566031470, −6.68073904860847667303907925149, −6.43696140097985091580915904063, −5.77646605717352910482039548748, −4.88570398264309540372289538731, −4.57377249545375000820952577892, −2.78879972867239885626681211306, −2.10657460298741432134768602087, −1.15455377293208093651449160211, 0, 1.15455377293208093651449160211, 2.10657460298741432134768602087, 2.78879972867239885626681211306, 4.57377249545375000820952577892, 4.88570398264309540372289538731, 5.77646605717352910482039548748, 6.43696140097985091580915904063, 6.68073904860847667303907925149, 7.56055034673341401976566031470

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