Properties

Label 2-4029-1.1-c1-0-72
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  − 1.59·2-s + 3-s + 0.548·4-s − 4.15·5-s − 1.59·6-s − 4.26·7-s + 2.31·8-s + 9-s + 6.63·10-s − 3.96·11-s + 0.548·12-s + 5.96·13-s + 6.80·14-s − 4.15·15-s − 4.79·16-s + 17-s − 1.59·18-s − 3.39·19-s − 2.27·20-s − 4.26·21-s + 6.32·22-s − 1.77·23-s + 2.31·24-s + 12.2·25-s − 9.52·26-s + 27-s − 2.33·28-s + ⋯
L(s)  = 1  − 1.12·2-s + 0.577·3-s + 0.274·4-s − 1.85·5-s − 0.651·6-s − 1.61·7-s + 0.819·8-s + 0.333·9-s + 2.09·10-s − 1.19·11-s + 0.158·12-s + 1.65·13-s + 1.81·14-s − 1.07·15-s − 1.19·16-s + 0.242·17-s − 0.376·18-s − 0.778·19-s − 0.509·20-s − 0.930·21-s + 1.34·22-s − 0.369·23-s + 0.473·24-s + 2.45·25-s − 1.86·26-s + 0.192·27-s − 0.441·28-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 \( 1 + 1.59T + 2T^{2} \)
5 \( 1 + 4.15T + 5T^{2} \)
7 \( 1 + 4.26T + 7T^{2} \)
11 \( 1 + 3.96T + 11T^{2} \)
13 \( 1 - 5.96T + 13T^{2} \)
19 \( 1 + 3.39T + 19T^{2} \)
23 \( 1 + 1.77T + 23T^{2} \)
29 \( 1 + 1.91T + 29T^{2} \)
31 \( 1 - 7.60T + 31T^{2} \)
37 \( 1 - 4.78T + 37T^{2} \)
41 \( 1 - 0.393T + 41T^{2} \)
43 \( 1 + 9.34T + 43T^{2} \)
47 \( 1 - 3.81T + 47T^{2} \)
53 \( 1 + 6.91T + 53T^{2} \)
59 \( 1 - 7.28T + 59T^{2} \)
61 \( 1 + 3.33T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 - 3.83T + 73T^{2} \)
83 \( 1 + 2.25T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + 2.41T + 97T^{2} \)
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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

−8.310690158649672457735814262627, −7.70663551498714019647622904819, −6.88425389965055606568915534401, −6.25745807673790245004452643618, −4.85698170253777769416989960209, −3.89271052492323125398536087817, −3.50907276406983638197952541271, −2.54770238688056858330348210569, −0.889809882043389140422425145954, 0, 0.889809882043389140422425145954, 2.54770238688056858330348210569, 3.50907276406983638197952541271, 3.89271052492323125398536087817, 4.85698170253777769416989960209, 6.25745807673790245004452643618, 6.88425389965055606568915534401, 7.70663551498714019647622904819, 8.310690158649672457735814262627

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