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Properties

Label 2-429-13.9-c1-0-2
Degree $2$
Conductor $429$
Sign $-0.697 - 0.716i$
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  + (−1.10 + 1.90i)2-s + (0.5 − 0.866i)3-s + (−1.42 − 2.46i)4-s − 0.484·5-s + (1.10 + 1.90i)6-s + (0.958 + 1.65i)7-s + 1.86·8-s + (−0.499 − 0.866i)9-s + (0.533 − 0.924i)10-s + (−0.5 + 0.866i)11-s − 2.84·12-s + (3.10 + 1.82i)13-s − 4.21·14-s + (−0.242 + 0.419i)15-s + (0.791 − 1.37i)16-s + (1.19 + 2.06i)17-s + ⋯
L(s)  = 1  + (−0.778 + 1.34i)2-s + (0.288 − 0.499i)3-s + (−0.712 − 1.23i)4-s − 0.216·5-s + (0.449 + 0.778i)6-s + (0.362 + 0.627i)7-s + 0.660·8-s + (−0.166 − 0.288i)9-s + (0.168 − 0.292i)10-s + (−0.150 + 0.261i)11-s − 0.822·12-s + (0.862 + 0.506i)13-s − 1.12·14-s + (−0.0626 + 0.108i)15-s + (0.197 − 0.342i)16-s + (0.288 + 0.500i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.697 - 0.716i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -0.697 - 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.344572 + 0.816341i\)
\(L(\frac12)\) \(\approx\) \(0.344572 + 0.816341i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-3.10 - 1.82i)T \)
good2 \( 1 + (1.10 - 1.90i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 0.484T + 5T^{2} \)
7 \( 1 + (-0.958 - 1.65i)T + (-3.5 + 6.06i)T^{2} \)
17 \( 1 + (-1.19 - 2.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.05 - 3.56i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.96 - 5.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.41 - 5.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + (4.72 - 8.18i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.666 + 1.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.11 + 5.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.23T + 47T^{2} \)
53 \( 1 - 8.17T + 53T^{2} \)
59 \( 1 + (6.00 + 10.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.74 + 3.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.80 - 3.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.81 - 4.88i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 + 5.62T + 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 + (0.222 - 0.386i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.32 - 7.49i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.66583616641077008069007075397, −10.20609293247449266188825990337, −9.349341426708033162880581822846, −8.352055394915318066504839351134, −8.027833211941989988690266943024, −6.98831044797071179565088943940, −6.10325146020739905792507310709, −5.27265958370672398706476599235, −3.55940496574291750265626916860, −1.63025682057704240754662340377, 0.75492536437589942541182002417, 2.44366595481948196939680086801, 3.53813201038946219161263581343, 4.47677766465174843401189597576, 6.01115792313565920218333343268, 7.66354754263717895710757469851, 8.343999299139601910217166546981, 9.251933326470478739906853492695, 10.11342495893412140187556214434, 10.72120703487553671266385417768

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