Properties

Label 2-4029-1.1-c1-0-200
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  + 2.10·2-s + 3-s + 2.41·4-s − 0.968·5-s + 2.10·6-s − 2.79·7-s + 0.874·8-s + 9-s − 2.03·10-s + 1.65·11-s + 2.41·12-s − 3.84·13-s − 5.87·14-s − 0.968·15-s − 2.99·16-s + 17-s + 2.10·18-s − 6.42·19-s − 2.33·20-s − 2.79·21-s + 3.47·22-s + 5.76·23-s + 0.874·24-s − 4.06·25-s − 8.08·26-s + 27-s − 6.75·28-s + ⋯
L(s)  = 1  + 1.48·2-s + 0.577·3-s + 1.20·4-s − 0.432·5-s + 0.857·6-s − 1.05·7-s + 0.309·8-s + 0.333·9-s − 0.643·10-s + 0.498·11-s + 0.697·12-s − 1.06·13-s − 1.56·14-s − 0.249·15-s − 0.748·16-s + 0.242·17-s + 0.495·18-s − 1.47·19-s − 0.523·20-s − 0.609·21-s + 0.740·22-s + 1.20·23-s + 0.178·24-s − 0.812·25-s − 1.58·26-s + 0.192·27-s − 1.27·28-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 \( 1 - 2.10T + 2T^{2} \)
5 \( 1 + 0.968T + 5T^{2} \)
7 \( 1 + 2.79T + 7T^{2} \)
11 \( 1 - 1.65T + 11T^{2} \)
13 \( 1 + 3.84T + 13T^{2} \)
19 \( 1 + 6.42T + 19T^{2} \)
23 \( 1 - 5.76T + 23T^{2} \)
29 \( 1 + 1.39T + 29T^{2} \)
31 \( 1 - 5.58T + 31T^{2} \)
37 \( 1 + 4.64T + 37T^{2} \)
41 \( 1 + 4.80T + 41T^{2} \)
43 \( 1 + 1.63T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 6.66T + 53T^{2} \)
59 \( 1 - 7.59T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + 16.3T + 73T^{2} \)
83 \( 1 - 3.58T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 1.40T + 97T^{2} \)
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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

−7.947547797213313807991247898633, −6.96971340406315118679290418382, −6.61964258842876210021689618764, −5.80047855126359073623897690988, −4.79588277141546883702913704144, −4.27641958162716356684584282296, −3.39749001865564538371758414071, −2.94486086206691247830369203434, −1.94655287168196823238305832513, 0, 1.94655287168196823238305832513, 2.94486086206691247830369203434, 3.39749001865564538371758414071, 4.27641958162716356684584282296, 4.79588277141546883702913704144, 5.80047855126359073623897690988, 6.61964258842876210021689618764, 6.96971340406315118679290418382, 7.947547797213313807991247898633

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