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Properties

Label 2-429-13.3-c1-0-21
Degree $2$
Conductor $429$
Sign $-0.729 - 0.684i$
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  + (−0.665 − 1.15i)2-s + (0.5 + 0.866i)3-s + (0.113 − 0.196i)4-s − 3.31·5-s + (0.665 − 1.15i)6-s + (0.767 − 1.32i)7-s − 2.96·8-s + (−0.499 + 0.866i)9-s + (2.20 + 3.82i)10-s + (0.5 + 0.866i)11-s + 0.226·12-s + (−1.88 − 3.07i)13-s − 2.04·14-s + (−1.65 − 2.86i)15-s + (1.74 + 3.02i)16-s + (−2.98 + 5.16i)17-s + ⋯
L(s)  = 1  + (−0.470 − 0.815i)2-s + (0.288 + 0.499i)3-s + (0.0565 − 0.0980i)4-s − 1.48·5-s + (0.271 − 0.470i)6-s + (0.289 − 0.502i)7-s − 1.04·8-s + (−0.166 + 0.288i)9-s + (0.697 + 1.20i)10-s + (0.150 + 0.261i)11-s + 0.0653·12-s + (−0.521 − 0.853i)13-s − 0.546·14-s + (−0.427 − 0.740i)15-s + (0.437 + 0.756i)16-s + (−0.723 + 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.729 - 0.684i$
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: \((2,\ 429,\ (\ :1/2),\ -0.729 - 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0332745 + 0.0840811i\)
\(L(\frac12)\) \(\approx\) \(0.0332745 + 0.0840811i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.88 + 3.07i)T \)
good2 \( 1 + (0.665 + 1.15i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 3.31T + 5T^{2} \)
7 \( 1 + (-0.767 + 1.32i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (2.98 - 5.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.23 - 5.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.66 + 4.60i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0956 - 0.165i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.38T + 31T^{2} \)
37 \( 1 + (4.63 + 8.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.125 + 0.216i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.58 + 4.47i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.36T + 47T^{2} \)
53 \( 1 - 7.13T + 53T^{2} \)
59 \( 1 + (-4.90 + 8.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.23 - 7.32i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.40 - 5.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.70 + 6.41i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 6.63T + 79T^{2} \)
83 \( 1 + 9.84T + 83T^{2} \)
89 \( 1 + (2.68 + 4.65i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.62 + 6.27i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.69640595936125117945110532540, −10.02885178057620023696229246255, −8.741923492820530134422826317654, −8.159677152338041984661573638257, −7.13983681542768191757515461424, −5.72258771299914019285780706601, −4.19378211890158878506461092268, −3.61581190800055504985717997641, −2.05199900807989137837843531349, −0.06040043974872681563940663812, 2.51152914093280371726106865926, 3.78806758039701526047467696194, 5.09734137495353386631706720707, 6.67184774889002573213287580655, 7.16907940904182272374017093018, 7.964553500915808370538249392031, 8.775448144436829942795446904135, 9.339043275392649175887659156926, 11.30744280507833956328232300792, 11.60868851862207476593640937799

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