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Properties

Label 2-429-143.109-c1-0-19
Degree $2$
Conductor $429$
Sign $-0.498 + 0.867i$
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  + (−0.200 − 0.200i)2-s − 3-s − 1.91i·4-s + (−0.906 + 0.906i)5-s + (0.200 + 0.200i)6-s + (2.29 − 2.29i)7-s + (−0.784 + 0.784i)8-s + 9-s + 0.363·10-s + (2.12 + 2.54i)11-s + 1.91i·12-s + (−0.213 − 3.59i)13-s − 0.919·14-s + (0.906 − 0.906i)15-s − 3.52·16-s − 2.75·17-s + ⋯
L(s)  = 1  + (−0.141 − 0.141i)2-s − 0.577·3-s − 0.959i·4-s + (−0.405 + 0.405i)5-s + (0.0817 + 0.0817i)6-s + (0.868 − 0.868i)7-s + (−0.277 + 0.277i)8-s + 0.333·9-s + 0.114·10-s + (0.641 + 0.767i)11-s + 0.554i·12-s + (−0.0592 − 0.998i)13-s − 0.245·14-s + (0.234 − 0.234i)15-s − 0.881·16-s − 0.667·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.498 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.498 + 0.867i$
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: \((2,\ 429,\ (\ :1/2),\ -0.498 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.433381 - 0.748706i\)
\(L(\frac12)\) \(\approx\) \(0.433381 - 0.748706i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 - 2.54i)T \)
13 \( 1 + (0.213 + 3.59i)T \)
good2 \( 1 + (0.200 + 0.200i)T + 2iT^{2} \)
5 \( 1 + (0.906 - 0.906i)T - 5iT^{2} \)
7 \( 1 + (-2.29 + 2.29i)T - 7iT^{2} \)
17 \( 1 + 2.75T + 17T^{2} \)
19 \( 1 + (5.56 + 5.56i)T + 19iT^{2} \)
23 \( 1 + 7.57iT - 23T^{2} \)
29 \( 1 + 7.96iT - 29T^{2} \)
31 \( 1 + (0.992 - 0.992i)T - 31iT^{2} \)
37 \( 1 + (-1.31 - 1.31i)T + 37iT^{2} \)
41 \( 1 + (-0.985 - 0.985i)T + 41iT^{2} \)
43 \( 1 - 5.76T + 43T^{2} \)
47 \( 1 + (1.14 + 1.14i)T + 47iT^{2} \)
53 \( 1 + 9.13T + 53T^{2} \)
59 \( 1 + (-4.44 - 4.44i)T + 59iT^{2} \)
61 \( 1 + 3.40iT - 61T^{2} \)
67 \( 1 + (-2.76 + 2.76i)T - 67iT^{2} \)
71 \( 1 + (-3.33 + 3.33i)T - 71iT^{2} \)
73 \( 1 + (-5.69 + 5.69i)T - 73iT^{2} \)
79 \( 1 - 5.17iT - 79T^{2} \)
83 \( 1 + (-6.70 - 6.70i)T + 83iT^{2} \)
89 \( 1 + (-3.99 - 3.99i)T + 89iT^{2} \)
97 \( 1 + (-12.2 + 12.2i)T - 97iT^{2} \)
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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

−10.94127425156355546389465453254, −10.27994107730520273341326047387, −9.232500611981515236617575224392, −8.043864536984570837674963404541, −6.95868034866111344126639328543, −6.27720691002254226951575231665, −4.85459449919709074310204811599, −4.29088583167637032590628766295, −2.22739846260301775739874646487, −0.62693444178508576714454119127, 1.89381702549032670382994343188, 3.71357870903593541658893374085, 4.58027053512892931697765505910, 5.83903738758843589362444689765, 6.81785604984481471460128240683, 7.977982352914337735592958493435, 8.646320387675449047979045694125, 9.321957023867224266053530149386, 10.96526188837438073304440914917, 11.55230691795597943749874564714

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