Properties

Label 2-429-1.1-c1-0-11
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·2-s + 3-s + 3.82·4-s − 3.41·5-s − 2.41·6-s + 0.828·7-s − 4.41·8-s + 9-s + 8.24·10-s + 11-s + 3.82·12-s − 13-s − 1.99·14-s − 3.41·15-s + 2.99·16-s − 2.58·17-s − 2.41·18-s − 6·19-s − 13.0·20-s + 0.828·21-s − 2.41·22-s + 4.82·23-s − 4.41·24-s + 6.65·25-s + 2.41·26-s + 27-s + 3.17·28-s + ⋯
L(s)  = 1  − 1.70·2-s + 0.577·3-s + 1.91·4-s − 1.52·5-s − 0.985·6-s + 0.313·7-s − 1.56·8-s + 0.333·9-s + 2.60·10-s + 0.301·11-s + 1.10·12-s − 0.277·13-s − 0.534·14-s − 0.881·15-s + 0.749·16-s − 0.627·17-s − 0.569·18-s − 1.37·19-s − 2.92·20-s + 0.180·21-s − 0.514·22-s + 1.00·23-s − 0.901·24-s + 1.33·25-s + 0.473·26-s + 0.192·27-s + 0.599·28-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: \(1\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
good2 \( 1 + 2.41T + 2T^{2} \)
5 \( 1 + 3.41T + 5T^{2} \)
7 \( 1 - 0.828T + 7T^{2} \)
17 \( 1 + 2.58T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 + 4.24T + 31T^{2} \)
37 \( 1 + 3.65T + 37T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + 9.07T + 43T^{2} \)
47 \( 1 - 8.48T + 47T^{2} \)
53 \( 1 + 9.31T + 53T^{2} \)
59 \( 1 + 5.17T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 + 4.48T + 73T^{2} \)
79 \( 1 - 5.07T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.73462947775384612173659269797, −9.572267351407206440797229255036, −8.625211138176998733528565617552, −8.294380618832128264164396604935, −7.31886910412356544374495004208, −6.73456846493927341009354436701, −4.63610094777111649562187410115, −3.39557814695746474405673163071, −1.85487998435817000029845974424, 0, 1.85487998435817000029845974424, 3.39557814695746474405673163071, 4.63610094777111649562187410115, 6.73456846493927341009354436701, 7.31886910412356544374495004208, 8.294380618832128264164396604935, 8.625211138176998733528565617552, 9.572267351407206440797229255036, 10.73462947775384612173659269797

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