Properties

Label 2-4011-1.1-c1-0-26
Degree $2$
Conductor $4011$
Sign $1$
Analytic cond. $32.0279$
Root an. cond. $5.65932$
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  − 0.163·2-s − 3-s − 1.97·4-s + 0.716·5-s + 0.163·6-s − 7-s + 0.651·8-s + 9-s − 0.117·10-s − 0.769·11-s + 1.97·12-s − 0.567·13-s + 0.163·14-s − 0.716·15-s + 3.83·16-s + 6.99·17-s − 0.163·18-s − 5.99·19-s − 1.41·20-s + 21-s + 0.126·22-s + 5.47·23-s − 0.651·24-s − 4.48·25-s + 0.0930·26-s − 27-s + 1.97·28-s + ⋯
L(s)  = 1  − 0.115·2-s − 0.577·3-s − 0.986·4-s + 0.320·5-s + 0.0669·6-s − 0.377·7-s + 0.230·8-s + 0.333·9-s − 0.0371·10-s − 0.232·11-s + 0.569·12-s − 0.157·13-s + 0.0438·14-s − 0.185·15-s + 0.959·16-s + 1.69·17-s − 0.0386·18-s − 1.37·19-s − 0.316·20-s + 0.218·21-s + 0.0269·22-s + 1.14·23-s − 0.132·24-s − 0.897·25-s + 0.0182·26-s − 0.192·27-s + 0.372·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $1$
Analytic conductor: \(32.0279\)
Root analytic conductor: \(5.65932\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4011,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8890707579\)
\(L(\frac12)\) \(\approx\) \(0.8890707579\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
191 \( 1 - T \)
good2 \( 1 + 0.163T + 2T^{2} \)
5 \( 1 - 0.716T + 5T^{2} \)
11 \( 1 + 0.769T + 11T^{2} \)
13 \( 1 + 0.567T + 13T^{2} \)
17 \( 1 - 6.99T + 17T^{2} \)
19 \( 1 + 5.99T + 19T^{2} \)
23 \( 1 - 5.47T + 23T^{2} \)
29 \( 1 - 2.95T + 29T^{2} \)
31 \( 1 + 7.76T + 31T^{2} \)
37 \( 1 + 4.22T + 37T^{2} \)
41 \( 1 - 3.27T + 41T^{2} \)
43 \( 1 - 6.03T + 43T^{2} \)
47 \( 1 + 5.82T + 47T^{2} \)
53 \( 1 + 4.99T + 53T^{2} \)
59 \( 1 + 1.13T + 59T^{2} \)
61 \( 1 + 6.81T + 61T^{2} \)
67 \( 1 + 1.23T + 67T^{2} \)
71 \( 1 - 9.20T + 71T^{2} \)
73 \( 1 - 15.7T + 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + 4.82T + 89T^{2} \)
97 \( 1 + 9.07T + 97T^{2} \)
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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

−8.442664153234608757298997821482, −7.78893227721280505921835007641, −6.98907615089150357192700959175, −6.03824794084827600545893964890, −5.48377401909569697679304628050, −4.80176216835580676221020904409, −3.91798740616613577069389489347, −3.11832359145386298824621032162, −1.74804046240218531962503877549, −0.57857053787115227855738535515, 0.57857053787115227855738535515, 1.74804046240218531962503877549, 3.11832359145386298824621032162, 3.91798740616613577069389489347, 4.80176216835580676221020904409, 5.48377401909569697679304628050, 6.03824794084827600545893964890, 6.98907615089150357192700959175, 7.78893227721280505921835007641, 8.442664153234608757298997821482

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