Properties

Label 2-4029-1.1-c1-0-129
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 3-s − 2.41·5-s − 1.41·6-s + 7-s + 2.82·8-s + 9-s + 3.41·10-s + 5.65·11-s − 1.41·14-s − 2.41·15-s − 4.00·16-s + 17-s − 1.41·18-s − 7.82·19-s + 21-s − 8.00·22-s − 7.24·23-s + 2.82·24-s + 0.828·25-s + 27-s − 0.828·29-s + 3.41·30-s + 9.65·31-s + ⋯
L(s)  = 1  − 1.00·2-s + 0.577·3-s − 1.07·5-s − 0.577·6-s + 0.377·7-s + 0.999·8-s + 0.333·9-s + 1.07·10-s + 1.70·11-s − 0.377·14-s − 0.623·15-s − 1.00·16-s + 0.242·17-s − 0.333·18-s − 1.79·19-s + 0.218·21-s − 1.70·22-s − 1.51·23-s + 0.577·24-s + 0.165·25-s + 0.192·27-s − 0.153·29-s + 0.623·30-s + 1.73·31-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: \(1\)
Selberg data: \((2,\ 4029,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
17 \( 1 - T \)
79 \( 1 + T \)
good2 \( 1 + 1.41T + 2T^{2} \)
5 \( 1 + 2.41T + 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
19 \( 1 + 7.82T + 19T^{2} \)
23 \( 1 + 7.24T + 23T^{2} \)
29 \( 1 + 0.828T + 29T^{2} \)
31 \( 1 - 9.65T + 31T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 + 3.65T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 - 4.07T + 47T^{2} \)
53 \( 1 + 9.58T + 53T^{2} \)
59 \( 1 + 6.41T + 59T^{2} \)
61 \( 1 - 10.6T + 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 - 1.07T + 71T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
83 \( 1 - 2.58T + 83T^{2} \)
89 \( 1 + 17.6T + 89T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−8.265395834249552639322132181131, −7.70292240234634113292200529227, −6.85193130662849459276157786156, −6.19323830807920932281814692863, −4.69498284059816075472090026346, −4.12908298892777774549381727150, −3.64957590584275959763457369839, −2.14834043955688921636054283977, −1.28838607629585746848245914722, 0, 1.28838607629585746848245914722, 2.14834043955688921636054283977, 3.64957590584275959763457369839, 4.12908298892777774549381727150, 4.69498284059816075472090026346, 6.19323830807920932281814692863, 6.85193130662849459276157786156, 7.70292240234634113292200529227, 8.265395834249552639322132181131

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