↑

Properties

Label 2-429-143.109-c1-0-11
Degree $2$
Conductor $429$
Sign $0.0377 + 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.63 − 1.63i)2-s + 3-s + 3.35i·4-s + (0.440 − 0.440i)5-s + (−1.63 − 1.63i)6-s + (2.71 − 2.71i)7-s + (2.22 − 2.22i)8-s + 9-s − 1.44·10-s + (1.99 + 2.64i)11-s + 3.35i·12-s + (2.23 + 2.82i)13-s − 8.89·14-s + (0.440 − 0.440i)15-s − 0.560·16-s − 3.11·17-s + ⋯
L(s)  = 1  + (−1.15 − 1.15i)2-s + 0.577·3-s + 1.67i·4-s + (0.196 − 0.196i)5-s + (−0.668 − 0.668i)6-s + (1.02 − 1.02i)7-s + (0.785 − 0.785i)8-s + 0.333·9-s − 0.455·10-s + (0.601 + 0.798i)11-s + 0.969i·12-s + (0.620 + 0.784i)13-s − 2.37·14-s + (0.113 − 0.113i)15-s − 0.140·16-s − 0.755·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0377 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0377 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.0377 + 0.999i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.0377 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802885 - 0.773114i\)
\(L(\frac12)\) \(\approx\) \(0.802885 - 0.773114i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 + (-1.99 - 2.64i)T \)
13 \( 1 + (-2.23 - 2.82i)T \)
good2 \( 1 + (1.63 + 1.63i)T + 2iT^{2} \)
5 \( 1 + (-0.440 + 0.440i)T - 5iT^{2} \)
7 \( 1 + (-2.71 + 2.71i)T - 7iT^{2} \)
17 \( 1 + 3.11T + 17T^{2} \)
19 \( 1 + (-2.98 - 2.98i)T + 19iT^{2} \)
23 \( 1 + 6.31iT - 23T^{2} \)
29 \( 1 + 4.54iT - 29T^{2} \)
31 \( 1 + (-5.18 + 5.18i)T - 31iT^{2} \)
37 \( 1 + (5.30 + 5.30i)T + 37iT^{2} \)
41 \( 1 + (-7.97 - 7.97i)T + 41iT^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (1.66 + 1.66i)T + 47iT^{2} \)
53 \( 1 + 8.11T + 53T^{2} \)
59 \( 1 + (0.774 + 0.774i)T + 59iT^{2} \)
61 \( 1 + 9.66iT - 61T^{2} \)
67 \( 1 + (-8.11 + 8.11i)T - 67iT^{2} \)
71 \( 1 + (6.86 - 6.86i)T - 71iT^{2} \)
73 \( 1 + (1.27 - 1.27i)T - 73iT^{2} \)
79 \( 1 - 4.85iT - 79T^{2} \)
83 \( 1 + (-7.42 - 7.42i)T + 83iT^{2} \)
89 \( 1 + (-1.31 - 1.31i)T + 89iT^{2} \)
97 \( 1 + (9.60 - 9.60i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.94387192627017202225821186444, −9.899439114358422265962666512170, −9.368530184560844541138174747096, −8.368722947501796778778861690717, −7.76024763341555888643278652491, −6.63212071189294118076108962005, −4.60507880367886640175039319798, −3.75153536891882530825340045725, −2.12778889547647120024528261818, −1.26761124525577675634035526560, 1.42968264355144972621821681330, 3.13988055586825823706046431718, 5.05658985381170127892484310899, 5.96606879808118983163370862792, 6.92147469080345271265557485161, 7.980908060203636091762223381768, 8.666811491758260381343909822628, 9.029281577871836406410395633391, 10.16915875244726822601575343641, 11.12812313116052496100395082929

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