Properties

Label 2-4029-1.1-c1-0-92
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  − 2.01·2-s + 3-s + 2.05·4-s − 0.898·5-s − 2.01·6-s − 4.86·7-s − 0.101·8-s + 9-s + 1.80·10-s − 0.654·11-s + 2.05·12-s − 4.49·13-s + 9.78·14-s − 0.898·15-s − 3.89·16-s − 17-s − 2.01·18-s + 4.80·19-s − 1.84·20-s − 4.86·21-s + 1.31·22-s + 7.89·23-s − 0.101·24-s − 4.19·25-s + 9.03·26-s + 27-s − 9.96·28-s + ⋯
L(s)  = 1  − 1.42·2-s + 0.577·3-s + 1.02·4-s − 0.401·5-s − 0.821·6-s − 1.83·7-s − 0.0359·8-s + 0.333·9-s + 0.572·10-s − 0.197·11-s + 0.591·12-s − 1.24·13-s + 2.61·14-s − 0.232·15-s − 0.974·16-s − 0.242·17-s − 0.474·18-s + 1.10·19-s − 0.412·20-s − 1.06·21-s + 0.280·22-s + 1.64·23-s − 0.0207·24-s − 0.838·25-s + 1.77·26-s + 0.192·27-s − 1.88·28-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 \( 1 + 2.01T + 2T^{2} \)
5 \( 1 + 0.898T + 5T^{2} \)
7 \( 1 + 4.86T + 7T^{2} \)
11 \( 1 + 0.654T + 11T^{2} \)
13 \( 1 + 4.49T + 13T^{2} \)
19 \( 1 - 4.80T + 19T^{2} \)
23 \( 1 - 7.89T + 23T^{2} \)
29 \( 1 - 2.96T + 29T^{2} \)
31 \( 1 - 6.47T + 31T^{2} \)
37 \( 1 - 9.04T + 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 - 3.33T + 47T^{2} \)
53 \( 1 + 4.23T + 53T^{2} \)
59 \( 1 - 5.44T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + 6.76T + 67T^{2} \)
71 \( 1 - 4.22T + 71T^{2} \)
73 \( 1 - 8.74T + 73T^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 - 1.07T + 89T^{2} \)
97 \( 1 - 6.15T + 97T^{2} \)
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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

−8.091947827332151418959892886131, −7.47676015569949652918759659223, −6.98243129689681568142699972106, −6.28158593188112967723612678005, −5.04850732038958243757560099634, −4.09245259112135979879805095032, −2.95388102314558527297743071023, −2.60527610113689175910138240936, −1.03754474500517492511232240094, 0, 1.03754474500517492511232240094, 2.60527610113689175910138240936, 2.95388102314558527297743071023, 4.09245259112135979879805095032, 5.04850732038958243757560099634, 6.28158593188112967723612678005, 6.98243129689681568142699972106, 7.47676015569949652918759659223, 8.091947827332151418959892886131

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