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Properties

Label 2-429-143.28-c1-0-25
Degree $2$
Conductor $429$
Sign $-0.861 + 0.507i$
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  + (1.14 − 0.0602i)2-s + (−0.104 + 0.994i)3-s + (−0.670 + 0.0704i)4-s + (−0.555 − 1.09i)5-s + (−0.0602 + 1.14i)6-s + (−3.71 − 3.00i)7-s + (−3.04 + 0.481i)8-s + (−0.978 − 0.207i)9-s + (−0.704 − 1.22i)10-s + (−2.85 − 1.68i)11-s − 0.674i·12-s + (1.18 + 3.40i)13-s + (−4.44 − 3.23i)14-s + (1.14 − 0.438i)15-s + (−2.14 + 0.456i)16-s + (2.10 + 2.34i)17-s + ⋯
L(s)  = 1  + (0.813 − 0.0426i)2-s + (−0.0603 + 0.574i)3-s + (−0.335 + 0.0352i)4-s + (−0.248 − 0.487i)5-s + (−0.0246 + 0.469i)6-s + (−1.40 − 1.13i)7-s + (−1.07 + 0.170i)8-s + (−0.326 − 0.0693i)9-s + (−0.222 − 0.386i)10-s + (−0.861 − 0.507i)11-s − 0.194i·12-s + (0.328 + 0.944i)13-s + (−1.18 − 0.864i)14-s + (0.295 − 0.113i)15-s + (−0.537 + 0.114i)16-s + (0.511 + 0.568i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.861 + 0.507i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.861 + 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0927463 - 0.340163i\)
\(L(\frac12)\) \(\approx\) \(0.0927463 - 0.340163i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.104 - 0.994i)T \)
11 \( 1 + (2.85 + 1.68i)T \)
13 \( 1 + (-1.18 - 3.40i)T \)
good2 \( 1 + (-1.14 + 0.0602i)T + (1.98 - 0.209i)T^{2} \)
5 \( 1 + (0.555 + 1.09i)T + (-2.93 + 4.04i)T^{2} \)
7 \( 1 + (3.71 + 3.00i)T + (1.45 + 6.84i)T^{2} \)
17 \( 1 + (-2.10 - 2.34i)T + (-1.77 + 16.9i)T^{2} \)
19 \( 1 + (4.85 + 1.86i)T + (14.1 + 12.7i)T^{2} \)
23 \( 1 + (1.49 - 0.860i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.786 + 1.76i)T + (-19.4 - 21.5i)T^{2} \)
31 \( 1 + (-4.40 - 2.24i)T + (18.2 + 25.0i)T^{2} \)
37 \( 1 + (2.15 + 5.61i)T + (-27.4 + 24.7i)T^{2} \)
41 \( 1 + (1.20 - 0.977i)T + (8.52 - 40.1i)T^{2} \)
43 \( 1 + (2.90 - 5.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.49 + 9.44i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (2.74 + 8.44i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.67 - 3.30i)T + (-12.2 - 57.7i)T^{2} \)
61 \( 1 + (8.45 - 7.61i)T + (6.37 - 60.6i)T^{2} \)
67 \( 1 + (10.2 + 2.73i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.565 + 0.0296i)T + (70.6 + 7.42i)T^{2} \)
73 \( 1 + (1.39 - 8.81i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (3.17 - 1.03i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-12.4 + 6.36i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + (-1.39 + 5.20i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (11.3 - 7.38i)T + (39.4 - 88.6i)T^{2} \)
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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

−10.66962066755836195819842900698, −10.01461156671733388564971009087, −9.020262347204580924390496275418, −8.224590532437911930483597469566, −6.71621555925347414098092729141, −5.90563592794304843203919026325, −4.62679883494158680690128137541, −3.97343358172580203107114820088, −3.06873800108714644750800173256, −0.16564622411155671084203878364, 2.71259379954323131593867367590, 3.32818173377413787883377910160, 4.94061308925461984013682723284, 5.92289333147322169420325207680, 6.50082827360415364505380144884, 7.79855070500452179001917337776, 8.795702184116182034951763020958, 9.753853623665864894451659871178, 10.64684811555528229500291625440, 12.09065691385369658449043763825

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