Properties

Label 2-6039-1.1-c1-0-33
Degree $2$
Conductor $6039$
Sign $1$
Analytic cond. $48.2216$
Root an. cond. $6.94418$
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  − 1.23·2-s − 0.470·4-s + 0.403·5-s − 0.303·7-s + 3.05·8-s − 0.498·10-s − 11-s − 3.53·13-s + 0.375·14-s − 2.83·16-s − 7.19·17-s + 3.67·19-s − 0.189·20-s + 1.23·22-s + 3.45·23-s − 4.83·25-s + 4.36·26-s + 0.142·28-s − 2.49·29-s + 2.86·31-s − 2.59·32-s + 8.90·34-s − 0.122·35-s + 3.00·37-s − 4.54·38-s + 1.23·40-s − 4.84·41-s + ⋯
L(s)  = 1  − 0.874·2-s − 0.235·4-s + 0.180·5-s − 0.114·7-s + 1.08·8-s − 0.157·10-s − 0.301·11-s − 0.979·13-s + 0.100·14-s − 0.709·16-s − 1.74·17-s + 0.842·19-s − 0.0424·20-s + 0.263·22-s + 0.720·23-s − 0.967·25-s + 0.856·26-s + 0.0269·28-s − 0.462·29-s + 0.515·31-s − 0.459·32-s + 1.52·34-s − 0.0206·35-s + 0.494·37-s − 0.736·38-s + 0.194·40-s − 0.757·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $1$
Analytic conductor: \(48.2216\)
Root analytic conductor: \(6.94418\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6039,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6196020736\)
\(L(\frac12)\) \(\approx\) \(0.6196020736\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + T \)
61 \( 1 - T \)
good2 \( 1 + 1.23T + 2T^{2} \)
5 \( 1 - 0.403T + 5T^{2} \)
7 \( 1 + 0.303T + 7T^{2} \)
13 \( 1 + 3.53T + 13T^{2} \)
17 \( 1 + 7.19T + 17T^{2} \)
19 \( 1 - 3.67T + 19T^{2} \)
23 \( 1 - 3.45T + 23T^{2} \)
29 \( 1 + 2.49T + 29T^{2} \)
31 \( 1 - 2.86T + 31T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 + 4.84T + 41T^{2} \)
43 \( 1 + 5.93T + 43T^{2} \)
47 \( 1 - 8.13T + 47T^{2} \)
53 \( 1 + 1.91T + 53T^{2} \)
59 \( 1 + 1.68T + 59T^{2} \)
67 \( 1 + 0.0898T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + 14.0T + 73T^{2} \)
79 \( 1 - 3.46T + 79T^{2} \)
83 \( 1 - 2.45T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + 1.70T + 97T^{2} \)
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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

−8.167566105605367522142820103916, −7.44409804697871708689097583397, −6.94218817284290201242883547722, −6.03794067814830876540913925238, −5.00806202883930693813710377091, −4.64987819132856487661707994009, −3.63248198676839949251262054953, −2.52987289213049555155706604006, −1.74032236886487742216909745881, −0.46765232997406625966849351772, 0.46765232997406625966849351772, 1.74032236886487742216909745881, 2.52987289213049555155706604006, 3.63248198676839949251262054953, 4.64987819132856487661707994009, 5.00806202883930693813710377091, 6.03794067814830876540913925238, 6.94218817284290201242883547722, 7.44409804697871708689097583397, 8.167566105605367522142820103916

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