Properties

Label 2-6014-1.1-c1-0-206
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 + 2.17·3-s + 4-s + 1.18·5-s − 2.17·6-s − 1.11·7-s − 8-s + 1.70·9-s − 1.18·10-s − 1.35·11-s + 2.17·12-s + 1.19·13-s + 1.11·14-s + 2.57·15-s + 16-s − 2.91·17-s − 1.70·18-s − 6.09·19-s + 1.18·20-s − 2.42·21-s + 1.35·22-s − 2.71·23-s − 2.17·24-s − 3.58·25-s − 1.19·26-s − 2.80·27-s − 1.11·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.25·3-s + 0.5·4-s + 0.531·5-s − 0.885·6-s − 0.422·7-s − 0.353·8-s + 0.569·9-s − 0.375·10-s − 0.408·11-s + 0.626·12-s + 0.332·13-s + 0.298·14-s + 0.665·15-s + 0.250·16-s − 0.707·17-s − 0.402·18-s − 1.39·19-s + 0.265·20-s − 0.529·21-s + 0.288·22-s − 0.565·23-s − 0.442·24-s − 0.717·25-s − 0.235·26-s − 0.538·27-s − 0.211·28-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 \( 1 - 2.17T + 3T^{2} \)
5 \( 1 - 1.18T + 5T^{2} \)
7 \( 1 + 1.11T + 7T^{2} \)
11 \( 1 + 1.35T + 11T^{2} \)
13 \( 1 - 1.19T + 13T^{2} \)
17 \( 1 + 2.91T + 17T^{2} \)
19 \( 1 + 6.09T + 19T^{2} \)
23 \( 1 + 2.71T + 23T^{2} \)
29 \( 1 - 6.09T + 29T^{2} \)
37 \( 1 - 8.60T + 37T^{2} \)
41 \( 1 - 8.96T + 41T^{2} \)
43 \( 1 - 3.76T + 43T^{2} \)
47 \( 1 + 9.20T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 - 9.22T + 59T^{2} \)
61 \( 1 + 5.54T + 61T^{2} \)
67 \( 1 - 6.43T + 67T^{2} \)
71 \( 1 + 1.35T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + 8.70T + 79T^{2} \)
83 \( 1 + 8.14T + 83T^{2} \)
89 \( 1 + 8.48T + 89T^{2} \)
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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

−8.059306491324967941336867971249, −7.23068245785569609642568911330, −6.24388975161861707344977727822, −6.03428153704683246625361729100, −4.63043701529300259119826697194, −3.89136524770810542107441597622, −2.83243732102453850266531270520, −2.41764910630393388419556826761, −1.54641747204695443503296857701, 0, 1.54641747204695443503296857701, 2.41764910630393388419556826761, 2.83243732102453850266531270520, 3.89136524770810542107441597622, 4.63043701529300259119826697194, 6.03428153704683246625361729100, 6.24388975161861707344977727822, 7.23068245785569609642568911330, 8.059306491324967941336867971249

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