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Properties

Label 2-429-429.428-c1-0-3
Degree $2$
Conductor $429$
Sign $0.986 + 0.166i$
Analytic cond. $3.42558$
Root an. cond. $1.85083$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.39i·2-s − 1.73·3-s − 3.73·4-s − 4.11·5-s + 4.14i·6-s + 4.14i·8-s + 2.99·9-s + 9.85i·10-s + (0.551 − 3.27i)11-s + 6.46·12-s + 3.60i·13-s + 7.12·15-s + 2.46·16-s − 7.18i·18-s + 15.3·20-s + ⋯
L(s)  = 1  − 1.69i·2-s − 1.00·3-s − 1.86·4-s − 1.84·5-s + 1.69i·6-s + 1.46i·8-s + 0.999·9-s + 3.11i·10-s + (0.166 − 0.986i)11-s + 1.86·12-s + 0.999i·13-s + 1.84·15-s + 0.616·16-s − 1.69i·18-s + 3.43·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.986 + 0.166i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.986 + 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.261536 - 0.0218859i\)
\(L(\frac12)\) \(\approx\) \(0.261536 - 0.0218859i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
11 \( 1 + (-0.551 + 3.27i)T \)
13 \( 1 - 3.60iT \)
good2 \( 1 + 2.39iT - 2T^{2} \)
5 \( 1 + 4.11T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 7.82iT - 41T^{2} \)
43 \( 1 - 12.4iT - 43T^{2} \)
47 \( 1 - 9.33T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 15.3T + 59T^{2} \)
61 \( 1 - 7.21iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 4.92T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 14.4iT - 79T^{2} \)
83 \( 1 - 12.6iT - 83T^{2} \)
89 \( 1 + 18.3T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.21358884841809274067381958812, −10.86680741271803203965132565918, −9.643826218707737760717214254835, −8.655634826082719970332304876766, −7.59590174923078312827299750737, −6.38199020432253896412392223338, −4.72442412879830577960760216072, −4.11937665389295349191829403198, −3.13991165025232550553253554097, −1.12874593625313832196527391657, 0.23725061969688850526275835641, 3.84254860659972554947160598838, 4.67620890011258919702863290808, 5.51843023201200654878721106054, 6.74699789152119159714439554773, 7.38203665988337009189483282479, 7.943630020148355366353831361896, 8.997471257011388623234500364843, 10.31121454121315419131673925015, 11.24538798715851420948817161875

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