↑

Properties

Label 2-429-143.32-c1-0-7
Degree $2$
Conductor $429$
Sign $0.533 - 0.846i$
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  + (0.225 − 0.0603i)2-s + (0.5 − 0.866i)3-s + (−1.68 + 0.972i)4-s + (−1.03 + 1.03i)5-s + (0.0603 − 0.225i)6-s + (1.55 + 0.415i)7-s + (−0.650 + 0.650i)8-s + (−0.499 − 0.866i)9-s + (−0.169 + 0.294i)10-s + (3.31 − 0.0413i)11-s + 1.94i·12-s + (−1.84 + 3.09i)13-s + 0.374·14-s + (0.377 + 1.40i)15-s + (1.83 − 3.18i)16-s + (3.85 + 6.67i)17-s + ⋯
L(s)  = 1  + (0.159 − 0.0426i)2-s + (0.288 − 0.499i)3-s + (−0.842 + 0.486i)4-s + (−0.461 + 0.461i)5-s + (0.0246 − 0.0919i)6-s + (0.585 + 0.157i)7-s + (−0.229 + 0.229i)8-s + (−0.166 − 0.288i)9-s + (−0.0537 + 0.0930i)10-s + (0.999 − 0.0124i)11-s + 0.561i·12-s + (−0.510 + 0.859i)13-s + 0.100·14-s + (0.0974 + 0.363i)15-s + (0.459 − 0.796i)16-s + (0.934 + 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.533 - 0.846i$
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: \((2,\ 429,\ (\ :1/2),\ 0.533 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10851 + 0.611823i\)
\(L(\frac12)\) \(\approx\) \(1.10851 + 0.611823i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 + 0.0413i)T \)
13 \( 1 + (1.84 - 3.09i)T \)
good2 \( 1 + (-0.225 + 0.0603i)T + (1.73 - i)T^{2} \)
5 \( 1 + (1.03 - 1.03i)T - 5iT^{2} \)
7 \( 1 + (-1.55 - 0.415i)T + (6.06 + 3.5i)T^{2} \)
17 \( 1 + (-3.85 - 6.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.30 - 4.85i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.19 + 0.687i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.26 - 1.30i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.16 - 2.16i)T - 31iT^{2} \)
37 \( 1 + (-6.44 + 1.72i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-8.17 + 2.18i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.90 + 10.2i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.06 + 8.06i)T + 47iT^{2} \)
53 \( 1 - 6.94T + 53T^{2} \)
59 \( 1 + (-0.876 + 3.27i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (8.07 - 4.65i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.34 + 5.01i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.47 - 1.19i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (6.00 - 6.00i)T - 73iT^{2} \)
79 \( 1 + 0.938iT - 79T^{2} \)
83 \( 1 + (-7.48 - 7.48i)T + 83iT^{2} \)
89 \( 1 + (-0.781 + 0.209i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (8.12 + 2.17i)T + (84.0 + 48.5i)T^{2} \)
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

−11.66042097188756711357225425448, −10.39660716161086597907521895703, −9.317112555589896588308642823779, −8.425984242743994215547225236659, −7.81426356987363034097724551703, −6.76792662277969634885948416383, −5.57568984732746097931332835951, −4.14826732126067270108994864689, −3.51707121347383831429344858005, −1.74654844692770388972514722574, 0.837092175123131319134471155905, 3.05174916803367556132950690095, 4.51153460803469078227611603950, 4.78139388422195707079462355462, 6.06268690053083472144799328728, 7.57743544891295029874654122837, 8.350836766307663596232079054409, 9.417326655439450954875017305030, 9.774204864533918071260004235660, 11.05063092926055432212065515200

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