Properties

Label 2-4029-1.1-c1-0-165
Degree $2$
Conductor $4029$
Sign $1$
Analytic cond. $32.1717$
Root an. cond. $5.67201$
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  + 2.09·2-s + 3-s + 2.36·4-s + 3.44·5-s + 2.09·6-s + 0.589·7-s + 0.770·8-s + 9-s + 7.19·10-s + 2.31·11-s + 2.36·12-s + 3.11·13-s + 1.23·14-s + 3.44·15-s − 3.12·16-s − 17-s + 2.09·18-s + 8.17·19-s + 8.15·20-s + 0.589·21-s + 4.83·22-s − 8.79·23-s + 0.770·24-s + 6.86·25-s + 6.51·26-s + 27-s + 1.39·28-s + ⋯
L(s)  = 1  + 1.47·2-s + 0.577·3-s + 1.18·4-s + 1.54·5-s + 0.853·6-s + 0.222·7-s + 0.272·8-s + 0.333·9-s + 2.27·10-s + 0.697·11-s + 0.683·12-s + 0.864·13-s + 0.329·14-s + 0.889·15-s − 0.781·16-s − 0.242·17-s + 0.492·18-s + 1.87·19-s + 1.82·20-s + 0.128·21-s + 1.03·22-s − 1.83·23-s + 0.157·24-s + 1.37·25-s + 1.27·26-s + 0.192·27-s + 0.263·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $1$
Analytic conductor: \(32.1717\)
Root analytic conductor: \(5.67201\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4029,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.657818260\)
\(L(\frac12)\) \(\approx\) \(7.657818260\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
17 \( 1 + T \)
79 \( 1 + T \)
good2 \( 1 - 2.09T + 2T^{2} \)
5 \( 1 - 3.44T + 5T^{2} \)
7 \( 1 - 0.589T + 7T^{2} \)
11 \( 1 - 2.31T + 11T^{2} \)
13 \( 1 - 3.11T + 13T^{2} \)
19 \( 1 - 8.17T + 19T^{2} \)
23 \( 1 + 8.79T + 23T^{2} \)
29 \( 1 - 1.82T + 29T^{2} \)
31 \( 1 + 7.98T + 31T^{2} \)
37 \( 1 + 1.76T + 37T^{2} \)
41 \( 1 + 7.42T + 41T^{2} \)
43 \( 1 + 6.21T + 43T^{2} \)
47 \( 1 + 0.237T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 - 9.49T + 59T^{2} \)
61 \( 1 + 8.69T + 61T^{2} \)
67 \( 1 + 1.10T + 67T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
83 \( 1 - 3.34T + 83T^{2} \)
89 \( 1 - 3.08T + 89T^{2} \)
97 \( 1 - 7.76T + 97T^{2} \)
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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

−8.600964520854270762747537641885, −7.45871612466287996389175353766, −6.65929217938902838453713925110, −5.97683347925044180688694862754, −5.51071039480527812636606322375, −4.73002514952335641130447817038, −3.72963310563838485533064493623, −3.22246744708477422882197399301, −2.10819955684818430192805736262, −1.51986090748815643726794148340, 1.51986090748815643726794148340, 2.10819955684818430192805736262, 3.22246744708477422882197399301, 3.72963310563838485533064493623, 4.73002514952335641130447817038, 5.51071039480527812636606322375, 5.97683347925044180688694862754, 6.65929217938902838453713925110, 7.45871612466287996389175353766, 8.600964520854270762747537641885

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