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Properties

Label 2-429-143.7-c1-0-21
Degree $2$
Conductor $429$
Sign $-0.485 + 0.874i$
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  + (0.701 − 1.82i)2-s + (−0.669 − 0.743i)3-s + (−1.36 − 1.22i)4-s + (2.43 + 0.384i)5-s + (−1.82 + 0.701i)6-s + (1.54 − 0.0808i)7-s + (0.288 − 0.147i)8-s + (−0.104 + 0.994i)9-s + (2.40 − 4.17i)10-s + (0.776 − 3.22i)11-s + 1.83i·12-s + (−0.441 + 3.57i)13-s + (0.934 − 2.87i)14-s + (−1.33 − 2.06i)15-s + (−0.449 − 4.28i)16-s + (1.13 + 0.504i)17-s + ⋯
L(s)  = 1  + (0.496 − 1.29i)2-s + (−0.386 − 0.429i)3-s + (−0.681 − 0.613i)4-s + (1.08 + 0.172i)5-s + (−0.746 + 0.286i)6-s + (0.583 − 0.0305i)7-s + (0.102 − 0.0520i)8-s + (−0.0348 + 0.331i)9-s + (0.761 − 1.31i)10-s + (0.233 − 0.972i)11-s + 0.529i·12-s + (−0.122 + 0.992i)13-s + (0.249 − 0.769i)14-s + (−0.345 − 0.532i)15-s + (−0.112 − 1.07i)16-s + (0.274 + 0.122i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.485 + 0.874i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.485 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02227 - 1.73795i\)
\(L(\frac12)\) \(\approx\) \(1.02227 - 1.73795i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.669 + 0.743i)T \)
11 \( 1 + (-0.776 + 3.22i)T \)
13 \( 1 + (0.441 - 3.57i)T \)
good2 \( 1 + (-0.701 + 1.82i)T + (-1.48 - 1.33i)T^{2} \)
5 \( 1 + (-2.43 - 0.384i)T + (4.75 + 1.54i)T^{2} \)
7 \( 1 + (-1.54 + 0.0808i)T + (6.96 - 0.731i)T^{2} \)
17 \( 1 + (-1.13 - 0.504i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (0.969 - 1.49i)T + (-7.72 - 17.3i)T^{2} \)
23 \( 1 + (3.90 + 2.25i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.884 - 4.16i)T + (-26.4 - 11.7i)T^{2} \)
31 \( 1 + (-0.00795 - 0.0502i)T + (-29.4 + 9.57i)T^{2} \)
37 \( 1 + (7.52 - 4.88i)T + (15.0 - 33.8i)T^{2} \)
41 \( 1 + (-0.828 - 0.0434i)T + (40.7 + 4.28i)T^{2} \)
43 \( 1 + (3.64 + 6.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.77 - 9.36i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-5.80 - 4.21i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.234 - 4.46i)T + (-58.6 + 6.16i)T^{2} \)
61 \( 1 + (-5.00 + 11.2i)T + (-40.8 - 45.3i)T^{2} \)
67 \( 1 + (-9.31 + 2.49i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.07 - 5.41i)T + (-52.7 + 47.5i)T^{2} \)
73 \( 1 + (-2.27 + 4.46i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (4.22 - 5.81i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.24 + 14.1i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (-2.43 - 9.08i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.12 - 4.95i)T + (20.1 - 94.8i)T^{2} \)
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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

−10.96273872928065844904008673394, −10.34338423372174598277779382874, −9.399817773614366865538697138403, −8.268700237512597854962284552704, −6.90870556180040807190329913758, −5.93869842089319758678229730795, −4.93334584174467687001501834499, −3.69655374947629998089517192539, −2.28887148575252847684153960057, −1.43157917999635758847688390868, 1.97064880706961337956896743061, 4.05213726959739225083381267965, 5.19498336939627597097878181308, 5.57022360872037844363182186814, 6.61043361174964890428196591718, 7.56053339794507838860269866619, 8.532633772709997686607041095627, 9.731673414345071334027877170166, 10.32235589642858552084316128616, 11.47601614410268772554692397344

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