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Properties

Label 2-429-11.5-c1-0-19
Degree $2$
Conductor $429$
Sign $-0.259 + 0.965i$
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  + (−0.375 − 1.15i)2-s + (0.809 + 0.587i)3-s + (0.423 − 0.308i)4-s + (0.339 − 1.04i)5-s + (0.375 − 1.15i)6-s + (0.237 − 0.172i)7-s + (−2.48 − 1.80i)8-s + (0.309 + 0.951i)9-s − 1.33·10-s + (−0.323 − 3.30i)11-s + 0.524·12-s + (−0.309 − 0.951i)13-s + (−0.288 − 0.209i)14-s + (0.890 − 0.646i)15-s + (−0.827 + 2.54i)16-s + (0.568 − 1.75i)17-s + ⋯
L(s)  = 1  + (−0.265 − 0.817i)2-s + (0.467 + 0.339i)3-s + (0.211 − 0.154i)4-s + (0.152 − 0.467i)5-s + (0.153 − 0.471i)6-s + (0.0896 − 0.0651i)7-s + (−0.877 − 0.637i)8-s + (0.103 + 0.317i)9-s − 0.422·10-s + (−0.0975 − 0.995i)11-s + 0.151·12-s + (−0.0857 − 0.263i)13-s + (−0.0769 − 0.0559i)14-s + (0.229 − 0.166i)15-s + (−0.206 + 0.636i)16-s + (0.137 − 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.259 + 0.965i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.259 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.924834 - 1.20577i\)
\(L(\frac12)\) \(\approx\) \(0.924834 - 1.20577i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.323 + 3.30i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (0.375 + 1.15i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.339 + 1.04i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-0.237 + 0.172i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (-0.568 + 1.75i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.351 - 0.255i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 2.58T + 23T^{2} \)
29 \( 1 + (-0.749 + 0.544i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.496 - 1.52i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.13 + 2.27i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.66 + 1.93i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.832T + 43T^{2} \)
47 \( 1 + (1.39 + 1.01i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.0466 - 0.143i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.09 + 2.24i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.19 - 6.74i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 2.04T + 67T^{2} \)
71 \( 1 + (4.11 - 12.6i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.68 + 1.22i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.63 - 5.02i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.251 + 0.772i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (-2.30 - 7.08i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.87200714392232176932985872430, −10.07119851005433598677563761013, −9.193302636774950533854630297067, −8.544663663630257192958765757374, −7.31800054212837900259192584407, −6.05626620451390361174486563572, −5.00931920755873023116183914013, −3.54371980602632725290374815106, −2.61315807780040320923821548101, −1.06180329020586573453631212529, 2.07963993833764807950707406998, 3.19703028091064035848937076526, 4.79380140480236029501954478198, 6.17483655185724224674434882497, 6.89885379149239975035009214558, 7.63102559786332920247826750524, 8.477123522170739082313094296227, 9.397106283524079616678817517223, 10.39737591797878134655941746157, 11.50695261358679828845098547791

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