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Properties

Label 2-429-13.10-c1-0-20
Degree $2$
Conductor $429$
Sign $0.843 + 0.537i$
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  + (2.15 − 1.24i)2-s + (0.5 + 0.866i)3-s + (2.10 − 3.64i)4-s + 1.90i·5-s + (2.15 + 1.24i)6-s + (1.37 + 0.795i)7-s − 5.50i·8-s + (−0.499 + 0.866i)9-s + (2.37 + 4.11i)10-s + (0.866 − 0.5i)11-s + 4.21·12-s + (−3.06 − 1.89i)13-s + 3.96·14-s + (−1.64 + 0.952i)15-s + (−2.65 − 4.59i)16-s + (−0.0999 + 0.173i)17-s + ⋯
L(s)  = 1  + (1.52 − 0.881i)2-s + (0.288 + 0.499i)3-s + (1.05 − 1.82i)4-s + 0.851i·5-s + (0.881 + 0.508i)6-s + (0.520 + 0.300i)7-s − 1.94i·8-s + (−0.166 + 0.288i)9-s + (0.750 + 1.30i)10-s + (0.261 − 0.150i)11-s + 1.21·12-s + (−0.849 − 0.526i)13-s + 1.05·14-s + (−0.425 + 0.245i)15-s + (−0.663 − 1.14i)16-s + (−0.0242 + 0.0419i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.843 + 0.537i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.843 + 0.537i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.25590 - 0.949805i\)
\(L(\frac12)\) \(\approx\) \(3.25590 - 0.949805i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (3.06 + 1.89i)T \)
good2 \( 1 + (-2.15 + 1.24i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 1.90iT - 5T^{2} \)
7 \( 1 + (-1.37 - 0.795i)T + (3.5 + 6.06i)T^{2} \)
17 \( 1 + (0.0999 - 0.173i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.94 + 3.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.40 - 2.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.04 + 7.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.50iT - 31T^{2} \)
37 \( 1 + (2.45 - 1.41i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.48 - 2.01i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.23 - 9.07i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.29iT - 47T^{2} \)
53 \( 1 - 9.92T + 53T^{2} \)
59 \( 1 + (-6.64 - 3.83i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.34 - 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.48 + 3.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.41 + 1.97i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.6iT - 73T^{2} \)
79 \( 1 + 0.250T + 79T^{2} \)
83 \( 1 + 0.221iT - 83T^{2} \)
89 \( 1 + (-8.44 + 4.87i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.72 + 3.30i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.29280622560824779059862914780, −10.50329359750038168116763702666, −9.758124247029494253663955466384, −8.413344044005800804365878807790, −7.05248942519309108890905390521, −5.97168182621904910848165341090, −4.98075660115390110960143886580, −4.10643005761688785382810511280, −2.98405463362775011512107019294, −2.18129174188202901790441402275, 1.97524534735113814511314451955, 3.61813188246915813845385873292, 4.63140470601971655197306955797, 5.30803872433347956924460965595, 6.58253272294387578822029861737, 7.20979965646156972399988989461, 8.235620238753066994031258432865, 8.995466584049408480738419385998, 10.57632098940652137484450605584, 11.85138683754706138164762028726

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