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Properties

Label 2-429-11.5-c1-0-2
Degree $2$
Conductor $429$
Sign $0.576 + 0.817i$
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  + (−0.839 − 2.58i)2-s + (−0.809 − 0.587i)3-s + (−4.35 + 3.16i)4-s + (0.0697 − 0.214i)5-s + (−0.839 + 2.58i)6-s + (−1.77 + 1.29i)7-s + (7.42 + 5.39i)8-s + (0.309 + 0.951i)9-s − 0.612·10-s + (2.74 + 1.86i)11-s + 5.37·12-s + (−0.309 − 0.951i)13-s + (4.82 + 3.50i)14-s + (−0.182 + 0.132i)15-s + (4.38 − 13.4i)16-s + (−1.17 + 3.62i)17-s + ⋯
L(s)  = 1  + (−0.593 − 1.82i)2-s + (−0.467 − 0.339i)3-s + (−2.17 + 1.58i)4-s + (0.0311 − 0.0959i)5-s + (−0.342 + 1.05i)6-s + (−0.671 + 0.487i)7-s + (2.62 + 1.90i)8-s + (0.103 + 0.317i)9-s − 0.193·10-s + (0.827 + 0.561i)11-s + 1.55·12-s + (−0.0857 − 0.263i)13-s + (1.28 + 0.937i)14-s + (−0.0471 + 0.0342i)15-s + (1.09 − 3.37i)16-s + (−0.285 + 0.880i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 + 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.576 + 0.817i$
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: \((2,\ 429,\ (\ :1/2),\ 0.576 + 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.540066 - 0.279995i\)
\(L(\frac12)\) \(\approx\) \(0.540066 - 0.279995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.74 - 1.86i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (0.839 + 2.58i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.0697 + 0.214i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (1.77 - 1.29i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (1.17 - 3.62i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.252 - 0.183i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.380T + 23T^{2} \)
29 \( 1 + (-2.40 + 1.74i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.02 + 3.16i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (9.39 - 6.82i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.48 - 5.43i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 5.84T + 43T^{2} \)
47 \( 1 + (0.229 + 0.166i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-4.07 - 12.5i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.82 + 1.32i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.17 + 9.75i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 7.53T + 67T^{2} \)
71 \( 1 + (1.52 - 4.69i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.78 + 3.47i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.49 - 10.7i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.04 + 9.36i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + (-0.804 - 2.47i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.03140506929977381990391305614, −10.27645720274243539873330769782, −9.408764393437748755130562810975, −8.773213558502653823199390191026, −7.64677221103284171744986446330, −6.29780309838270482474454842674, −4.83964578146210501922164279768, −3.72762635817567067548384359084, −2.51689670319176631086711898780, −1.24699274143498995262483889996, 0.60110665218307792077758628862, 3.81093857905641207967506898192, 4.91062681942219789402187412416, 5.88739269048731958578238863945, 6.80197574440987354138555775754, 7.21166852065763844117749282066, 8.677299897183315325727086520784, 9.170260819296123174082516273193, 10.10192083210185624598409283161, 10.88668898833148828358489361488

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