Properties

Label 2-4029-1.1-c1-0-169
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  + 0.785·2-s + 3-s − 1.38·4-s − 0.680·5-s + 0.785·6-s + 0.961·7-s − 2.65·8-s + 9-s − 0.534·10-s + 0.176·11-s − 1.38·12-s + 0.855·13-s + 0.755·14-s − 0.680·15-s + 0.677·16-s − 17-s + 0.785·18-s + 1.48·19-s + 0.941·20-s + 0.961·21-s + 0.138·22-s − 5.05·23-s − 2.65·24-s − 4.53·25-s + 0.672·26-s + 27-s − 1.33·28-s + ⋯
L(s)  = 1  + 0.555·2-s + 0.577·3-s − 0.691·4-s − 0.304·5-s + 0.320·6-s + 0.363·7-s − 0.939·8-s + 0.333·9-s − 0.169·10-s + 0.0531·11-s − 0.399·12-s + 0.237·13-s + 0.201·14-s − 0.175·15-s + 0.169·16-s − 0.242·17-s + 0.185·18-s + 0.339·19-s + 0.210·20-s + 0.209·21-s + 0.0295·22-s − 1.05·23-s − 0.542·24-s − 0.907·25-s + 0.131·26-s + 0.192·27-s − 0.251·28-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 \( 1 - 0.785T + 2T^{2} \)
5 \( 1 + 0.680T + 5T^{2} \)
7 \( 1 - 0.961T + 7T^{2} \)
11 \( 1 - 0.176T + 11T^{2} \)
13 \( 1 - 0.855T + 13T^{2} \)
19 \( 1 - 1.48T + 19T^{2} \)
23 \( 1 + 5.05T + 23T^{2} \)
29 \( 1 + 1.15T + 29T^{2} \)
31 \( 1 + 7.75T + 31T^{2} \)
37 \( 1 - 9.51T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 - 1.31T + 43T^{2} \)
47 \( 1 - 3.37T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 - 8.92T + 59T^{2} \)
61 \( 1 - 5.96T + 61T^{2} \)
67 \( 1 + 1.68T + 67T^{2} \)
71 \( 1 + 3.12T + 71T^{2} \)
73 \( 1 + 0.346T + 73T^{2} \)
83 \( 1 + 0.884T + 83T^{2} \)
89 \( 1 + 8.32T + 89T^{2} \)
97 \( 1 - 1.42T + 97T^{2} \)
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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

−8.122768079615600061969438441211, −7.54456958724031048002690698446, −6.51056230044799105975764678399, −5.69452223822955782448298626225, −4.98437717733183217601286403411, −4.05219341441867199973950559641, −3.71496307612372660039193752697, −2.66670684717488258986147442875, −1.54935454630965385113105187025, 0, 1.54935454630965385113105187025, 2.66670684717488258986147442875, 3.71496307612372660039193752697, 4.05219341441867199973950559641, 4.98437717733183217601286403411, 5.69452223822955782448298626225, 6.51056230044799105975764678399, 7.54456958724031048002690698446, 8.122768079615600061969438441211

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