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Properties

Label 2-429-143.32-c1-0-8
Degree $2$
Conductor $429$
Sign $0.925 + 0.379i$
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  + (−2.55 + 0.684i)2-s + (−0.5 + 0.866i)3-s + (4.32 − 2.49i)4-s + (−2.81 + 2.81i)5-s + (0.684 − 2.55i)6-s + (−2.71 − 0.727i)7-s + (−5.60 + 5.60i)8-s + (−0.499 − 0.866i)9-s + (5.26 − 9.12i)10-s + (0.108 − 3.31i)11-s + 4.99i·12-s + (−1.12 + 3.42i)13-s + 7.43·14-s + (−1.03 − 3.84i)15-s + (5.48 − 9.49i)16-s + (0.408 + 0.707i)17-s + ⋯
L(s)  = 1  + (−1.80 + 0.484i)2-s + (−0.288 + 0.499i)3-s + (2.16 − 1.24i)4-s + (−1.25 + 1.25i)5-s + (0.279 − 1.04i)6-s + (−1.02 − 0.274i)7-s + (−1.98 + 1.98i)8-s + (−0.166 − 0.288i)9-s + (1.66 − 2.88i)10-s + (0.0328 − 0.999i)11-s + 1.44i·12-s + (−0.312 + 0.949i)13-s + 1.98·14-s + (−0.266 − 0.993i)15-s + (1.37 − 2.37i)16-s + (0.0990 + 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.925 + 0.379i$
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: \((2,\ 429,\ (\ :1/2),\ 0.925 + 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152131 - 0.0299835i\)
\(L(\frac12)\) \(\approx\) \(0.152131 - 0.0299835i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.108 + 3.31i)T \)
13 \( 1 + (1.12 - 3.42i)T \)
good2 \( 1 + (2.55 - 0.684i)T + (1.73 - i)T^{2} \)
5 \( 1 + (2.81 - 2.81i)T - 5iT^{2} \)
7 \( 1 + (2.71 + 0.727i)T + (6.06 + 3.5i)T^{2} \)
17 \( 1 + (-0.408 - 0.707i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0308 - 0.115i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.97 + 3.45i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.48 - 2.58i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.79 - 4.79i)T - 31iT^{2} \)
37 \( 1 + (-8.78 + 2.35i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.35 + 1.97i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.75 - 3.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.67 - 1.67i)T + 47iT^{2} \)
53 \( 1 - 6.49T + 53T^{2} \)
59 \( 1 + (-2.54 + 9.49i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.23 + 0.713i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.73 + 6.46i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.79 - 0.750i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.797 + 0.797i)T - 73iT^{2} \)
79 \( 1 + 7.81iT - 79T^{2} \)
83 \( 1 + (10.4 + 10.4i)T + 83iT^{2} \)
89 \( 1 + (9.16 - 2.45i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.41 - 1.18i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.83833186646458604112298611132, −10.19945790979643398572869472587, −9.333510536853209173201071755013, −8.371625637390743181488769176517, −7.49373766845027261748057160428, −6.65861172548676450232028688670, −6.12418400256454026045596865838, −3.93549600770883823102888124412, −2.76761952653151388410545431374, −0.24601115523013791605681021421, 0.906893679019553083224571391701, 2.55718493128933650139440983887, 4.06075372584890593688221250184, 5.79866950530160926242066337732, 7.19700603583080877499119892555, 7.76092035278085325866808484252, 8.461908099087186564385449351799, 9.487776693760423872008487879842, 9.993029476861738461438502700856, 11.26969739020194837530241267433

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