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Properties

Label 2-429-13.3-c1-0-6
Degree $2$
Conductor $429$
Sign $-0.951 + 0.307i$
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  + (1.24 + 2.14i)2-s + (0.5 + 0.866i)3-s + (−2.07 + 3.60i)4-s − 1.71·5-s + (−1.24 + 2.14i)6-s + (−1.04 + 1.81i)7-s − 5.35·8-s + (−0.499 + 0.866i)9-s + (−2.12 − 3.68i)10-s + (−0.5 − 0.866i)11-s − 4.15·12-s + (3.60 − 0.160i)13-s − 5.20·14-s + (−0.857 − 1.48i)15-s + (−2.48 − 4.30i)16-s + (4.06 − 7.03i)17-s + ⋯
L(s)  = 1  + (0.877 + 1.51i)2-s + (0.288 + 0.499i)3-s + (−1.03 + 1.80i)4-s − 0.766·5-s + (−0.506 + 0.877i)6-s + (−0.396 + 0.686i)7-s − 1.89·8-s + (−0.166 + 0.288i)9-s + (−0.672 − 1.16i)10-s + (−0.150 − 0.261i)11-s − 1.20·12-s + (0.999 − 0.0445i)13-s − 1.39·14-s + (−0.221 − 0.383i)15-s + (−0.621 − 1.07i)16-s + (0.985 − 1.70i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.951 + 0.307i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.951 + 0.307i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.281758 - 1.78718i\)
\(L(\frac12)\) \(\approx\) \(0.281758 - 1.78718i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-3.60 + 0.160i)T \)
good2 \( 1 + (-1.24 - 2.14i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 1.71T + 5T^{2} \)
7 \( 1 + (1.04 - 1.81i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (-4.06 + 7.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.28 - 3.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.30 - 5.72i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.63 - 6.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.14T + 31T^{2} \)
37 \( 1 + (1.64 + 2.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.93 - 3.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.653 + 1.13i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.30T + 47T^{2} \)
53 \( 1 - 13.5T + 53T^{2} \)
59 \( 1 + (-2.61 + 4.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.56 + 4.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.35 + 2.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.29 + 5.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.00T + 73T^{2} \)
79 \( 1 - 2.75T + 79T^{2} \)
83 \( 1 - 1.20T + 83T^{2} \)
89 \( 1 + (-0.0810 - 0.140i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.00 + 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.94600980742901754317507939497, −10.88638629412319617910270233732, −9.414932891529623418448147051892, −8.674779999636479460550474759468, −7.78075369859348627911699255133, −7.05104488962565990875108480338, −5.75033532537913355831229610994, −5.23693519652238826301498632709, −3.86984393419106844848578595600, −3.23770210371004814119404739671, 0.906385381463413925406852524691, 2.39108932555824451048395854590, 3.74511965149375735470144621824, 4.08557180848119645692427635366, 5.63684851291080263085613487419, 6.80310080010643336464754824138, 8.036000050906163523536397808779, 9.009313550313948704134147770111, 10.34862076628618186349124091174, 10.71412178192156964674601637504

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