Properties

Label 2-4029-1.1-c1-0-142
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  − 1.99·2-s + 3-s + 1.97·4-s + 1.24·5-s − 1.99·6-s − 4.06·7-s + 0.0495·8-s + 9-s − 2.47·10-s + 3.04·11-s + 1.97·12-s + 2.14·13-s + 8.10·14-s + 1.24·15-s − 4.04·16-s + 17-s − 1.99·18-s + 2.62·19-s + 2.44·20-s − 4.06·21-s − 6.06·22-s + 0.657·23-s + 0.0495·24-s − 3.46·25-s − 4.27·26-s + 27-s − 8.02·28-s + ⋯
L(s)  = 1  − 1.40·2-s + 0.577·3-s + 0.987·4-s + 0.554·5-s − 0.813·6-s − 1.53·7-s + 0.0175·8-s + 0.333·9-s − 0.781·10-s + 0.916·11-s + 0.570·12-s + 0.594·13-s + 2.16·14-s + 0.320·15-s − 1.01·16-s + 0.242·17-s − 0.469·18-s + 0.603·19-s + 0.547·20-s − 0.886·21-s − 1.29·22-s + 0.137·23-s + 0.0101·24-s − 0.692·25-s − 0.838·26-s + 0.192·27-s − 1.51·28-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: \(1\)
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 \( 1 + 1.99T + 2T^{2} \)
5 \( 1 - 1.24T + 5T^{2} \)
7 \( 1 + 4.06T + 7T^{2} \)
11 \( 1 - 3.04T + 11T^{2} \)
13 \( 1 - 2.14T + 13T^{2} \)
19 \( 1 - 2.62T + 19T^{2} \)
23 \( 1 - 0.657T + 23T^{2} \)
29 \( 1 + 6.92T + 29T^{2} \)
31 \( 1 + 9.42T + 31T^{2} \)
37 \( 1 - 0.281T + 37T^{2} \)
41 \( 1 + 5.67T + 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 + 0.718T + 47T^{2} \)
53 \( 1 - 3.71T + 53T^{2} \)
59 \( 1 + 7.57T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + 7.51T + 67T^{2} \)
71 \( 1 - 4.69T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
83 \( 1 + 2.47T + 83T^{2} \)
89 \( 1 - 5.34T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−8.285987800028965326923463860863, −7.43157044386731522234198231654, −6.82683788746232512044722674454, −6.21737107713128078443054581611, −5.28658356601524113483126995071, −3.80829160327354912018305773327, −3.38848375600911246708820300457, −2.12554711210968698813220049734, −1.35020668113318652333428413764, 0, 1.35020668113318652333428413764, 2.12554711210968698813220049734, 3.38848375600911246708820300457, 3.80829160327354912018305773327, 5.28658356601524113483126995071, 6.21737107713128078443054581611, 6.82683788746232512044722674454, 7.43157044386731522234198231654, 8.285987800028965326923463860863

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