Properties

Label 2-4011-1.1-c1-0-41
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  + 2.22·2-s − 3-s + 2.95·4-s − 3.65·5-s − 2.22·6-s + 7-s + 2.12·8-s + 9-s − 8.14·10-s − 5.04·11-s − 2.95·12-s + 3.81·13-s + 2.22·14-s + 3.65·15-s − 1.17·16-s − 0.994·17-s + 2.22·18-s + 5.71·19-s − 10.8·20-s − 21-s − 11.2·22-s − 2.03·23-s − 2.12·24-s + 8.37·25-s + 8.48·26-s − 27-s + 2.95·28-s + ⋯
L(s)  = 1  + 1.57·2-s − 0.577·3-s + 1.47·4-s − 1.63·5-s − 0.908·6-s + 0.377·7-s + 0.752·8-s + 0.333·9-s − 2.57·10-s − 1.52·11-s − 0.853·12-s + 1.05·13-s + 0.594·14-s + 0.944·15-s − 0.293·16-s − 0.241·17-s + 0.524·18-s + 1.31·19-s − 2.41·20-s − 0.218·21-s − 2.39·22-s − 0.423·23-s − 0.434·24-s + 1.67·25-s + 1.66·26-s − 0.192·27-s + 0.558·28-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

\(L(1)\) \(\approx\) \(2.515062120\)
\(L(\frac12)\) \(\approx\) \(2.515062120\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.22T + 2T^{2} \)
5 \( 1 + 3.65T + 5T^{2} \)
11 \( 1 + 5.04T + 11T^{2} \)
13 \( 1 - 3.81T + 13T^{2} \)
17 \( 1 + 0.994T + 17T^{2} \)
19 \( 1 - 5.71T + 19T^{2} \)
23 \( 1 + 2.03T + 23T^{2} \)
29 \( 1 - 0.807T + 29T^{2} \)
31 \( 1 - 1.21T + 31T^{2} \)
37 \( 1 + 1.13T + 37T^{2} \)
41 \( 1 - 0.234T + 41T^{2} \)
43 \( 1 - 6.74T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 4.70T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 2.43T + 67T^{2} \)
71 \( 1 + 5.80T + 71T^{2} \)
73 \( 1 + 0.0792T + 73T^{2} \)
79 \( 1 - 7.06T + 79T^{2} \)
83 \( 1 + 4.00T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−8.104953483061807478708330762148, −7.54378658277298552888846776515, −6.91898053873343207079042012278, −5.89659508378136399512481015178, −5.32760877515672001922052635416, −4.65789645644323719844389861950, −3.95577757541188660038272323252, −3.35940331216820098701255100602, −2.41507446834935598862022086120, −0.72295959140129791470079058178, 0.72295959140129791470079058178, 2.41507446834935598862022086120, 3.35940331216820098701255100602, 3.95577757541188660038272323252, 4.65789645644323719844389861950, 5.32760877515672001922052635416, 5.89659508378136399512481015178, 6.91898053873343207079042012278, 7.54378658277298552888846776515, 8.104953483061807478708330762148

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