Properties

Label 2-429-1.1-c1-0-16
Degree $2$
Conductor $429$
Sign $1$
Analytic cond. $3.42558$
Root an. cond. $1.85083$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.67·2-s + 3-s + 5.15·4-s − 0.481·5-s + 2.67·6-s − 4.15·7-s + 8.44·8-s + 9-s − 1.28·10-s − 11-s + 5.15·12-s − 13-s − 11.1·14-s − 0.481·15-s + 12.2·16-s − 1.67·17-s + 2.67·18-s + 4.15·19-s − 2.48·20-s − 4.15·21-s − 2.67·22-s − 5.50·23-s + 8.44·24-s − 4.76·25-s − 2.67·26-s + 27-s − 21.4·28-s + ⋯
L(s)  = 1  + 1.89·2-s + 0.577·3-s + 2.57·4-s − 0.215·5-s + 1.09·6-s − 1.57·7-s + 2.98·8-s + 0.333·9-s − 0.407·10-s − 0.301·11-s + 1.48·12-s − 0.277·13-s − 2.97·14-s − 0.124·15-s + 3.06·16-s − 0.406·17-s + 0.630·18-s + 0.953·19-s − 0.554·20-s − 0.906·21-s − 0.570·22-s − 1.14·23-s + 1.72·24-s − 0.953·25-s − 0.524·26-s + 0.192·27-s − 4.05·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.119410879\)
\(L(\frac12)\) \(\approx\) \(4.119410879\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
good2 \( 1 - 2.67T + 2T^{2} \)
5 \( 1 + 0.481T + 5T^{2} \)
7 \( 1 + 4.15T + 7T^{2} \)
17 \( 1 + 1.67T + 17T^{2} \)
19 \( 1 - 4.15T + 19T^{2} \)
23 \( 1 + 5.50T + 23T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 - 6.48T + 31T^{2} \)
37 \( 1 + 6.70T + 37T^{2} \)
41 \( 1 - 2.15T + 41T^{2} \)
43 \( 1 + 7.86T + 43T^{2} \)
47 \( 1 - 8.57T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 - 7.66T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 9.47T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 + 1.75T + 79T^{2} \)
83 \( 1 - 8.96T + 83T^{2} \)
89 \( 1 - 2.48T + 89T^{2} \)
97 \( 1 - 2.70T + 97T^{2} \)
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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

−11.67551806700917807474034922967, −10.34621603135489014254872045491, −9.669412849485241661081748318746, −8.088314656115432470468388245891, −7.02457230757647359030767263560, −6.33825554731416592035776507122, −5.30762276865224137516316276676, −4.06050500485667345506197370127, −3.30994060828330222802108136608, −2.37502782587245148335031686945, 2.37502782587245148335031686945, 3.30994060828330222802108136608, 4.06050500485667345506197370127, 5.30762276865224137516316276676, 6.33825554731416592035776507122, 7.02457230757647359030767263560, 8.088314656115432470468388245891, 9.669412849485241661081748318746, 10.34621603135489014254872045491, 11.67551806700917807474034922967

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