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Properties

Label 2-429-11.3-c1-0-0
Degree $2$
Conductor $429$
Sign $-0.117 - 0.993i$
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  + (0.788 − 0.573i)2-s + (−0.309 − 0.951i)3-s + (−0.324 + 0.997i)4-s + (−2.91 − 2.11i)5-s + (−0.788 − 0.573i)6-s + (−1.06 + 3.28i)7-s + (0.918 + 2.82i)8-s + (−0.809 + 0.587i)9-s − 3.51·10-s + (0.466 + 3.28i)11-s + 1.04·12-s + (−0.809 + 0.587i)13-s + (1.03 + 3.20i)14-s + (−1.11 + 3.42i)15-s + (0.648 + 0.470i)16-s + (−6.51 − 4.73i)17-s + ⋯
L(s)  = 1  + (0.557 − 0.405i)2-s + (−0.178 − 0.549i)3-s + (−0.162 + 0.498i)4-s + (−1.30 − 0.946i)5-s + (−0.322 − 0.234i)6-s + (−0.403 + 1.24i)7-s + (0.324 + 0.999i)8-s + (−0.269 + 0.195i)9-s − 1.10·10-s + (0.140 + 0.990i)11-s + 0.302·12-s + (−0.224 + 0.163i)13-s + (0.277 + 0.855i)14-s + (−0.287 + 0.883i)15-s + (0.162 + 0.117i)16-s + (−1.58 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.117 - 0.993i$
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: \((2,\ 429,\ (\ :1/2),\ -0.117 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.418638 + 0.470911i\)
\(L(\frac12)\) \(\approx\) \(0.418638 + 0.470911i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.466 - 3.28i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.788 + 0.573i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (2.91 + 2.11i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.06 - 3.28i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (6.51 + 4.73i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.40 - 4.30i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.38T + 23T^{2} \)
29 \( 1 + (0.565 - 1.73i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.04 - 4.39i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.848 + 2.61i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.68 + 8.25i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.68T + 43T^{2} \)
47 \( 1 + (-0.462 - 1.42i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.30 - 4.57i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.57 - 4.85i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.09 + 0.796i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.00T + 67T^{2} \)
71 \( 1 + (-7.39 - 5.37i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.38 + 7.35i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.72 + 1.97i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.69 - 3.41i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 + (0.0735 - 0.0534i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.79324119256742839663170469441, −11.05779250250479371936761651784, −9.156264854876360742555316214330, −8.806712637933569845338684035435, −7.70434392962495597084897883865, −6.93507474065823114524721447962, −5.25609539513102801684173283098, −4.65039181924906753123734234595, −3.43261853023103765984845427604, −2.13017387225665398303428987200, 0.33488895623735117904622929105, 3.33994797783007065535296753094, 3.99689073163757873880799707418, 4.90028112873373235452350400306, 6.44948840743788429696156766504, 6.83346325809694858147881874482, 7.956407413457889048692814840953, 9.203432123285018746486711871743, 10.30870725183988819818629618489, 11.05051381383605176099386159880

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