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Properties

Label 2-429-143.32-c1-0-0
Degree $2$
Conductor $429$
Sign $-0.980 - 0.195i$
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  + (−2.44 + 0.654i)2-s + (0.5 − 0.866i)3-s + (3.81 − 2.20i)4-s + (−0.748 + 0.748i)5-s + (−0.654 + 2.44i)6-s + (−0.270 − 0.0724i)7-s + (−4.29 + 4.29i)8-s + (−0.499 − 0.866i)9-s + (1.33 − 2.31i)10-s + (−2.59 + 2.07i)11-s − 4.40i·12-s + (−2.42 − 2.66i)13-s + 0.708·14-s + (0.273 + 1.02i)15-s + (3.28 − 5.69i)16-s + (0.0934 + 0.161i)17-s + ⋯
L(s)  = 1  + (−1.72 + 0.463i)2-s + (0.288 − 0.499i)3-s + (1.90 − 1.10i)4-s + (−0.334 + 0.334i)5-s + (−0.267 + 0.997i)6-s + (−0.102 − 0.0273i)7-s + (−1.51 + 1.51i)8-s + (−0.166 − 0.288i)9-s + (0.423 − 0.733i)10-s + (−0.781 + 0.624i)11-s − 1.27i·12-s + (−0.673 − 0.739i)13-s + 0.189·14-s + (0.0707 + 0.263i)15-s + (0.821 − 1.42i)16-s + (0.0226 + 0.0392i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.980 - 0.195i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.980 - 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00967825 + 0.0979990i\)
\(L(\frac12)\) \(\approx\) \(0.00967825 + 0.0979990i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (2.59 - 2.07i)T \)
13 \( 1 + (2.42 + 2.66i)T \)
good2 \( 1 + (2.44 - 0.654i)T + (1.73 - i)T^{2} \)
5 \( 1 + (0.748 - 0.748i)T - 5iT^{2} \)
7 \( 1 + (0.270 + 0.0724i)T + (6.06 + 3.5i)T^{2} \)
17 \( 1 + (-0.0934 - 0.161i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.228 - 0.851i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.93 + 1.11i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.15 - 1.24i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.46 - 6.46i)T - 31iT^{2} \)
37 \( 1 + (-4.61 + 1.23i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (11.4 - 3.07i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.00 + 1.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.49 + 2.49i)T + 47iT^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + (0.394 - 1.47i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.06 - 3.50i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.153 - 0.572i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.46 - 0.660i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.86 - 3.86i)T - 73iT^{2} \)
79 \( 1 - 9.45iT - 79T^{2} \)
83 \( 1 + (0.0514 + 0.0514i)T + 83iT^{2} \)
89 \( 1 + (-12.9 + 3.46i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.320 - 0.0858i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.22151385839385071174054519634, −10.34975917145320567561366090969, −9.760132564815226687763468960733, −8.695822086832794440362446546732, −7.88548394492288064502143716560, −7.32827414264132997526142046846, −6.53244852975672361343535315385, −5.22022651376135871886230904669, −3.07863134606424961218281119560, −1.76690613377142073917282682771, 0.098610309087472964082584360131, 2.06355807405126949980484788315, 3.24827767304291957864052841408, 4.73803080865389240015501844770, 6.33893154321580185310415454002, 7.63313335436576902009002656235, 8.153220626129050890456239166229, 9.085279390092717063311472881627, 9.713244828531039416852091463049, 10.52053883930387177721251846325

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