Properties

Label 2-6011-1.1-c1-0-485
Degree $2$
Conductor $6011$
Sign $-1$
Analytic cond. $47.9980$
Root an. cond. $6.92806$
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  + 1.41·2-s + 2·3-s + 2.82·5-s + 2.82·6-s − 7-s − 2.82·8-s + 9-s + 4.00·10-s − 4.24·11-s − 3.41·13-s − 1.41·14-s + 5.65·15-s − 4.00·16-s − 1.41·17-s + 1.41·18-s + 2·19-s − 2·21-s − 6·22-s − 4.82·23-s − 5.65·24-s + 3.00·25-s − 4.82·26-s − 4·27-s + 8.48·29-s + 8.00·30-s + ⋯
L(s)  = 1  + 1.00·2-s + 1.15·3-s + 1.26·5-s + 1.15·6-s − 0.377·7-s − 0.999·8-s + 0.333·9-s + 1.26·10-s − 1.27·11-s − 0.946·13-s − 0.377·14-s + 1.46·15-s − 1.00·16-s − 0.342·17-s + 0.333·18-s + 0.458·19-s − 0.436·21-s − 1.27·22-s − 1.00·23-s − 1.15·24-s + 0.600·25-s − 0.946·26-s − 0.769·27-s + 1.57·29-s + 1.46·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6011\)
Sign: $-1$
Analytic conductor: \(47.9980\)
Root analytic conductor: \(6.92806\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6011,\ (\ :1/2),\ -1)\)

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)$
bad6011 \( 1+O(T) \)
good2 \( 1 - 1.41T + 2T^{2} \)
3 \( 1 - 2T + 3T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 + 8.24T + 31T^{2} \)
37 \( 1 + 8.24T + 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + 2.58T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 3.34T + 59T^{2} \)
61 \( 1 - 4.82T + 61T^{2} \)
67 \( 1 - 1.34T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 1.48T + 97T^{2} \)
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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

−7.80227777642978981045443275430, −6.89131336795170439683812530891, −6.12411223570190490749209900801, −5.33599932400916277552699035183, −5.02718628241489753749585940691, −3.92453793980422204078593130716, −3.13490711696784360345618452452, −2.52817507160489180094504075009, −1.97859821165285673217133941978, 0, 1.97859821165285673217133941978, 2.52817507160489180094504075009, 3.13490711696784360345618452452, 3.92453793980422204078593130716, 5.02718628241489753749585940691, 5.33599932400916277552699035183, 6.12411223570190490749209900801, 6.89131336795170439683812530891, 7.80227777642978981045443275430

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