Properties

Label 2-4029-1.1-c1-0-97
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  − 0.107·2-s + 3-s − 1.98·4-s − 4.11·5-s − 0.107·6-s − 2.41·7-s + 0.428·8-s + 9-s + 0.441·10-s + 3.22·11-s − 1.98·12-s + 1.21·13-s + 0.260·14-s − 4.11·15-s + 3.93·16-s − 17-s − 0.107·18-s − 3.26·19-s + 8.17·20-s − 2.41·21-s − 0.346·22-s − 0.375·23-s + 0.428·24-s + 11.8·25-s − 0.130·26-s + 27-s + 4.80·28-s + ⋯
L(s)  = 1  − 0.0760·2-s + 0.577·3-s − 0.994·4-s − 1.83·5-s − 0.0438·6-s − 0.914·7-s + 0.151·8-s + 0.333·9-s + 0.139·10-s + 0.971·11-s − 0.574·12-s + 0.336·13-s + 0.0694·14-s − 1.06·15-s + 0.982·16-s − 0.242·17-s − 0.0253·18-s − 0.748·19-s + 1.82·20-s − 0.527·21-s − 0.0738·22-s − 0.0783·23-s + 0.0875·24-s + 2.37·25-s − 0.0255·26-s + 0.192·27-s + 0.908·28-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 \( 1 + 0.107T + 2T^{2} \)
5 \( 1 + 4.11T + 5T^{2} \)
7 \( 1 + 2.41T + 7T^{2} \)
11 \( 1 - 3.22T + 11T^{2} \)
13 \( 1 - 1.21T + 13T^{2} \)
19 \( 1 + 3.26T + 19T^{2} \)
23 \( 1 + 0.375T + 23T^{2} \)
29 \( 1 + 2.01T + 29T^{2} \)
31 \( 1 - 6.34T + 31T^{2} \)
37 \( 1 - 8.38T + 37T^{2} \)
41 \( 1 + 5.02T + 41T^{2} \)
43 \( 1 - 8.14T + 43T^{2} \)
47 \( 1 - 0.107T + 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 + 5.52T + 59T^{2} \)
61 \( 1 + 9.40T + 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 - 8.17T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
83 \( 1 + 1.92T + 83T^{2} \)
89 \( 1 + 9.78T + 89T^{2} \)
97 \( 1 - 7.20T + 97T^{2} \)
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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

−8.138970113855934157619332373019, −7.56637126642976114528275354078, −6.75797707695031027630718106424, −5.97930195424929831353276983035, −4.56373409553872453062497672195, −4.19041218727973862554614901307, −3.60101252413551080357993676227, −2.84685567204689012062249470074, −1.07189289662706409509893950424, 0, 1.07189289662706409509893950424, 2.84685567204689012062249470074, 3.60101252413551080357993676227, 4.19041218727973862554614901307, 4.56373409553872453062497672195, 5.97930195424929831353276983035, 6.75797707695031027630718106424, 7.56637126642976114528275354078, 8.138970113855934157619332373019

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