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Properties

Label 2-429-11.3-c1-0-17
Degree $2$
Conductor $429$
Sign $-0.313 + 0.949i$
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  + (−0.297 + 0.216i)2-s + (−0.309 − 0.951i)3-s + (−0.576 + 1.77i)4-s + (−0.859 − 0.624i)5-s + (0.297 + 0.216i)6-s + (1.08 − 3.35i)7-s + (−0.439 − 1.35i)8-s + (−0.809 + 0.587i)9-s + 0.391·10-s + (−0.965 + 3.17i)11-s + 1.86·12-s + (0.809 − 0.587i)13-s + (0.401 + 1.23i)14-s + (−0.328 + 1.01i)15-s + (−2.59 − 1.88i)16-s + (−3.20 − 2.32i)17-s + ⋯
L(s)  = 1  + (−0.210 + 0.153i)2-s + (−0.178 − 0.549i)3-s + (−0.288 + 0.886i)4-s + (−0.384 − 0.279i)5-s + (0.121 + 0.0883i)6-s + (0.411 − 1.26i)7-s + (−0.155 − 0.478i)8-s + (−0.269 + 0.195i)9-s + 0.123·10-s + (−0.291 + 0.956i)11-s + 0.538·12-s + (0.224 − 0.163i)13-s + (0.107 + 0.329i)14-s + (−0.0847 + 0.260i)15-s + (−0.648 − 0.470i)16-s + (−0.777 − 0.564i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.313 + 0.949i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.313 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396476 - 0.548556i\)
\(L(\frac12)\) \(\approx\) \(0.396476 - 0.548556i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.965 - 3.17i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (0.297 - 0.216i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (0.859 + 0.624i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.08 + 3.35i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (3.20 + 2.32i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.45 + 7.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.09T + 23T^{2} \)
29 \( 1 + (-0.949 + 2.92i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.78 + 4.20i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.875 - 2.69i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.82 + 8.70i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.48T + 43T^{2} \)
47 \( 1 + (-1.79 - 5.51i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.40 - 4.65i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.15 - 6.61i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.52 - 2.56i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 + (10.0 + 7.30i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.03 - 6.26i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.94 - 5.04i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.82 - 7.13i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 5.44T + 89T^{2} \)
97 \( 1 + (-3.26 + 2.37i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.04305796872565969630940435532, −9.973983216419232324447905670466, −8.830265169662373936435232062991, −7.988067188674888380257947396757, −7.28226360727176698831517404457, −6.62919930233312747817197641543, −4.69724249871461881321274185736, −4.21763258789100271440866265644, −2.53861530802825477585110396343, −0.46841628696141568771114145380, 1.86376570139355683522117911285, 3.44213172124223374046462098506, 4.79979027524425288016020585648, 5.72838821693458520559726607461, 6.38102828069228330131344414152, 8.326087931216101327331901040751, 8.596129228037830260172540661264, 9.738938131161391140287037818972, 10.57733231520664134836444544561, 11.27212247666033166571001652273

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