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Properties

Label 2-429-11.5-c1-0-8
Degree $2$
Conductor $429$
Sign $-0.986 + 0.161i$
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  + (0.711 + 2.18i)2-s + (0.809 + 0.587i)3-s + (−2.66 + 1.93i)4-s + (−0.951 + 2.92i)5-s + (−0.711 + 2.18i)6-s + (−0.0471 + 0.0342i)7-s + (−2.41 − 1.75i)8-s + (0.309 + 0.951i)9-s − 7.08·10-s + (2.87 − 1.65i)11-s − 3.29·12-s + (−0.309 − 0.951i)13-s + (−0.108 − 0.0788i)14-s + (−2.49 + 1.80i)15-s + (0.0823 − 0.253i)16-s + (1.21 − 3.74i)17-s + ⋯
L(s)  = 1  + (0.502 + 1.54i)2-s + (0.467 + 0.339i)3-s + (−1.33 + 0.968i)4-s + (−0.425 + 1.30i)5-s + (−0.290 + 0.893i)6-s + (−0.0178 + 0.0129i)7-s + (−0.852 − 0.619i)8-s + (0.103 + 0.317i)9-s − 2.23·10-s + (0.866 − 0.498i)11-s − 0.951·12-s + (−0.0857 − 0.263i)13-s + (−0.0289 − 0.0210i)14-s + (−0.642 + 0.467i)15-s + (0.0205 − 0.0633i)16-s + (0.294 − 0.907i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.986 + 0.161i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.986 + 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.150001 - 1.84456i\)
\(L(\frac12)\) \(\approx\) \(0.150001 - 1.84456i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-2.87 + 1.65i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (-0.711 - 2.18i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.951 - 2.92i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (0.0471 - 0.0342i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (-1.21 + 3.74i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.87 + 1.36i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 2.01T + 23T^{2} \)
29 \( 1 + (-0.807 + 0.586i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.329 - 1.01i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.39 - 3.19i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.00 - 4.36i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.88T + 43T^{2} \)
47 \( 1 + (-9.32 - 6.77i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.14 + 9.66i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.63 + 5.54i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.782 - 2.40i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 + (1.42 - 4.37i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-8.18 + 5.94i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.35 + 13.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.166 + 0.512i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 9.66T + 89T^{2} \)
97 \( 1 + (1.08 + 3.33i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.53874563975455700873948716623, −10.77269815507781667270209464568, −9.575697587432824558936422890988, −8.595728028702429430579881362275, −7.69608557963947730583324666357, −6.92921039683770119563086361444, −6.25899391140740235891562721752, −5.02027599877779039420377623166, −3.89754528470414516122182921002, −2.93215500980004825641049886737, 1.07503679439944496319618126559, 2.12466176716172386508387338221, 3.74554866458603359613410019580, 4.28780567961037663714380017897, 5.44966988447657446649972300399, 7.02079172705965007831144959975, 8.346207669782057061089785504412, 9.040544011816707876969966336939, 9.850395195897393530477718889383, 10.88318706410022119278423268669

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