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Properties

Label 2-429-429.233-c1-0-27
Degree $2$
Conductor $429$
Sign $0.992 - 0.124i$
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  + (0.658 + 0.213i)2-s + (1.40 − 1.01i)3-s + (−1.23 − 0.893i)4-s + (1.34 + 4.14i)5-s + (1.14 − 0.370i)6-s + (−1.43 − 1.97i)8-s + (0.927 − 2.85i)9-s + 3.01i·10-s + (3.28 − 0.486i)11-s − 2.63·12-s + (3.42 + 1.11i)13-s + (6.10 + 4.43i)15-s + (0.418 + 1.28i)16-s + (1.22 − 1.68i)18-s + (2.04 − 6.30i)20-s + ⋯
L(s)  = 1  + (0.465 + 0.151i)2-s + (0.809 − 0.587i)3-s + (−0.615 − 0.446i)4-s + (0.602 + 1.85i)5-s + (0.465 − 0.151i)6-s + (−0.506 − 0.697i)8-s + (0.309 − 0.951i)9-s + 0.954i·10-s + (0.989 − 0.146i)11-s − 0.760·12-s + (0.951 + 0.309i)13-s + (1.57 + 1.14i)15-s + (0.104 + 0.321i)16-s + (0.287 − 0.396i)18-s + (0.457 − 1.40i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.992 - 0.124i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.992 - 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21167 + 0.138630i\)
\(L(\frac12)\) \(\approx\) \(2.21167 + 0.138630i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.40 + 1.01i)T \)
11 \( 1 + (-3.28 + 0.486i)T \)
13 \( 1 + (-3.42 - 1.11i)T \)
good2 \( 1 + (-0.658 - 0.213i)T + (1.61 + 1.17i)T^{2} \)
5 \( 1 + (-1.34 - 4.14i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.16 + 6.65i)T^{2} \)
17 \( 1 + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.59 + 6.32i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 12.4iT - 43T^{2} \)
47 \( 1 + (10.0 - 7.31i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.47 - 2.52i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (14.6 - 4.76i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + (-1.52 - 4.68i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-5.16 - 1.67i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.35 + 0.764i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 18.3T + 89T^{2} \)
97 \( 1 + (78.4 + 57.0i)T^{2} \)
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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

−11.12673808711597540593905964916, −10.17127778846879278588431513157, −9.400493813403062238256957279725, −8.542906688099101223265742260803, −7.11346300872838184560219956254, −6.52900870648694023858936893746, −5.81360376632574608173296808323, −3.92044741070455245617142706909, −3.22204878140359757386810690056, −1.74563327943393086088621931121, 1.54936099776255617383310033042, 3.33361391940575832875911127700, 4.38101257881199707712192655737, 4.93075367109869918245127923752, 6.06626487240851779985359899718, 8.061829601986365339107065011157, 8.499943951716984138183295495260, 9.317381820574854850622024535938, 9.752618980304394374879214213658, 11.31535275617728840724917712317

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