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Properties

Label 2-429-11.9-c1-0-10
Degree $2$
Conductor $429$
Sign $0.868 - 0.496i$
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  + (0.0279 − 0.0858i)2-s + (0.809 − 0.587i)3-s + (1.61 + 1.17i)4-s + (0.964 + 2.96i)5-s + (−0.0279 − 0.0858i)6-s + (−2.21 − 1.61i)7-s + (0.291 − 0.211i)8-s + (0.309 − 0.951i)9-s + 0.281·10-s + (3.27 − 0.527i)11-s + 1.99·12-s + (0.309 − 0.951i)13-s + (−0.200 + 0.145i)14-s + (2.52 + 1.83i)15-s + (1.22 + 3.75i)16-s + (0.167 + 0.514i)17-s + ⋯
L(s)  = 1  + (0.0197 − 0.0607i)2-s + (0.467 − 0.339i)3-s + (0.805 + 0.585i)4-s + (0.431 + 1.32i)5-s + (−0.0113 − 0.0350i)6-s + (−0.838 − 0.609i)7-s + (0.103 − 0.0749i)8-s + (0.103 − 0.317i)9-s + 0.0890·10-s + (0.987 − 0.158i)11-s + 0.574·12-s + (0.0857 − 0.263i)13-s + (−0.0535 + 0.0389i)14-s + (0.651 + 0.473i)15-s + (0.305 + 0.939i)16-s + (0.0405 + 0.124i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.868 - 0.496i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.868 - 0.496i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90889 + 0.507356i\)
\(L(\frac12)\) \(\approx\) \(1.90889 + 0.507356i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.27 + 0.527i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (-0.0279 + 0.0858i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.964 - 2.96i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.21 + 1.61i)T + (2.16 + 6.65i)T^{2} \)
17 \( 1 + (-0.167 - 0.514i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.41 - 2.48i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 5.16T + 23T^{2} \)
29 \( 1 + (3.17 + 2.30i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.23 - 6.86i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (7.09 + 5.15i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.64 + 1.92i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.39T + 43T^{2} \)
47 \( 1 + (-2.17 + 1.58i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.91 + 8.98i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.83 + 2.05i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.67 + 11.3i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 4.08T + 67T^{2} \)
71 \( 1 + (2.06 + 6.35i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.11 + 5.89i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.429 + 1.32i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.92 - 15.1i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 17.2T + 89T^{2} \)
97 \( 1 + (-4.53 + 13.9i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.00338196377741594351649408597, −10.57116708787873262702828292088, −9.518951706891658386559576821924, −8.406342547663045325231224372605, −7.12232324823436134756257364291, −6.86756144651445147342381835570, −6.02006394353344975968012658510, −3.72225319686596978814947604272, −3.20016866493249572896794802501, −1.94813951142514087378528308925, 1.42702985802910525423099408462, 2.73872004682165839614889687829, 4.29854646799949189153232441207, 5.40909072677751288819238969505, 6.26700307698674780695707575714, 7.26980999977898984050572543102, 8.855442020396322041217119341616, 9.125841097989370128476814160563, 9.956772505989906595150723298921, 11.07727344766356222994413664095

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