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Properties

Label 2-429-143.32-c1-0-6
Degree $2$
Conductor $429$
Sign $-0.229 - 0.973i$
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  + (1.43 − 0.385i)2-s + (−0.5 + 0.866i)3-s + (0.185 − 0.107i)4-s + (−1.55 + 1.55i)5-s + (−0.385 + 1.43i)6-s + (2.05 + 0.551i)7-s + (−1.87 + 1.87i)8-s + (−0.499 − 0.866i)9-s + (−1.63 + 2.83i)10-s + (−3.31 − 0.109i)11-s + 0.214i·12-s + (1.07 + 3.44i)13-s + 3.17·14-s + (−0.568 − 2.12i)15-s + (−2.19 + 3.79i)16-s + (−1.88 − 3.27i)17-s + ⋯
L(s)  = 1  + (1.01 − 0.272i)2-s + (−0.288 + 0.499i)3-s + (0.0929 − 0.0536i)4-s + (−0.694 + 0.694i)5-s + (−0.157 + 0.586i)6-s + (0.778 + 0.208i)7-s + (−0.664 + 0.664i)8-s + (−0.166 − 0.288i)9-s + (−0.516 + 0.894i)10-s + (−0.999 − 0.0330i)11-s + 0.0619i·12-s + (0.298 + 0.954i)13-s + 0.848·14-s + (−0.146 − 0.547i)15-s + (−0.547 + 0.948i)16-s + (−0.457 − 0.793i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.229 - 0.973i$
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: \((2,\ 429,\ (\ :1/2),\ -0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.913611 + 1.15434i\)
\(L(\frac12)\) \(\approx\) \(0.913611 + 1.15434i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.31 + 0.109i)T \)
13 \( 1 + (-1.07 - 3.44i)T \)
good2 \( 1 + (-1.43 + 0.385i)T + (1.73 - i)T^{2} \)
5 \( 1 + (1.55 - 1.55i)T - 5iT^{2} \)
7 \( 1 + (-2.05 - 0.551i)T + (6.06 + 3.5i)T^{2} \)
17 \( 1 + (1.88 + 3.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.33 - 4.98i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.66 - 2.11i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.17 - 1.83i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.23 + 6.23i)T - 31iT^{2} \)
37 \( 1 + (-4.04 + 1.08i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.728 - 0.195i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.35 - 4.08i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.03 + 5.03i)T + 47iT^{2} \)
53 \( 1 - 8.86T + 53T^{2} \)
59 \( 1 + (-1.80 + 6.74i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.01 - 0.586i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0671 - 0.250i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.596 - 0.159i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.07 + 3.07i)T - 73iT^{2} \)
79 \( 1 - 8.01iT - 79T^{2} \)
83 \( 1 + (-6.52 - 6.52i)T + 83iT^{2} \)
89 \( 1 + (13.2 - 3.56i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.58 + 0.959i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.38672448408052317679165432954, −11.05072144071710916859141095503, −9.798163588153330352190539781838, −8.613313623951118199449421090109, −7.77574015057444046523569187248, −6.50098980490195296793620746325, −5.33554832438521205430317008426, −4.56063791102966349883373644719, −3.64366240230811308054398318788, −2.49601029529584365262667018322, 0.71775217668672385951849952846, 2.87699829953945028428537108118, 4.46069347882294029811629428367, 4.89348232745562772590268171842, 5.94595487281213510423166954561, 7.03304853053297990407633850113, 8.155243238080768109801857213979, 8.677052661929355510700628104317, 10.28758887080186340948310445637, 11.10680746065131838736457687461

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