Properties

Label 2-429-143.21-c1-0-9
Degree $2$
Conductor $429$
Sign $0.993 + 0.112i$
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  + (1.06 − 1.06i)2-s − 3-s − 0.261i·4-s + (0.340 + 0.340i)5-s + (−1.06 + 1.06i)6-s + (0.593 + 0.593i)7-s + (1.84 + 1.84i)8-s + 9-s + 0.724·10-s + (2.67 + 1.96i)11-s + 0.261i·12-s + (−2.47 + 2.61i)13-s + 1.26·14-s + (−0.340 − 0.340i)15-s + 4.45·16-s + 0.904·17-s + ⋯
L(s)  = 1  + (0.751 − 0.751i)2-s − 0.577·3-s − 0.130i·4-s + (0.152 + 0.152i)5-s + (−0.434 + 0.434i)6-s + (0.224 + 0.224i)7-s + (0.653 + 0.653i)8-s + 0.333·9-s + 0.229·10-s + (0.806 + 0.591i)11-s + 0.0755i·12-s + (−0.687 + 0.726i)13-s + 0.337·14-s + (−0.0880 − 0.0880i)15-s + 1.11·16-s + 0.219·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.993 + 0.112i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88125 - 0.106421i\)
\(L(\frac12)\) \(\approx\) \(1.88125 - 0.106421i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.67 - 1.96i)T \)
13 \( 1 + (2.47 - 2.61i)T \)
good2 \( 1 + (-1.06 + 1.06i)T - 2iT^{2} \)
5 \( 1 + (-0.340 - 0.340i)T + 5iT^{2} \)
7 \( 1 + (-0.593 - 0.593i)T + 7iT^{2} \)
17 \( 1 - 0.904T + 17T^{2} \)
19 \( 1 + (-2.48 + 2.48i)T - 19iT^{2} \)
23 \( 1 + 4.23iT - 23T^{2} \)
29 \( 1 - 3.83iT - 29T^{2} \)
31 \( 1 + (-5.60 - 5.60i)T + 31iT^{2} \)
37 \( 1 + (-5.73 + 5.73i)T - 37iT^{2} \)
41 \( 1 + (2.91 - 2.91i)T - 41iT^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 + (5.97 - 5.97i)T - 47iT^{2} \)
53 \( 1 - 3.19T + 53T^{2} \)
59 \( 1 + (5.71 - 5.71i)T - 59iT^{2} \)
61 \( 1 + 3.56iT - 61T^{2} \)
67 \( 1 + (10.0 + 10.0i)T + 67iT^{2} \)
71 \( 1 + (10.7 + 10.7i)T + 71iT^{2} \)
73 \( 1 + (-10.3 - 10.3i)T + 73iT^{2} \)
79 \( 1 + 7.59iT - 79T^{2} \)
83 \( 1 + (5.02 - 5.02i)T - 83iT^{2} \)
89 \( 1 + (-3.53 + 3.53i)T - 89iT^{2} \)
97 \( 1 + (-0.0569 - 0.0569i)T + 97iT^{2} \)
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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

−11.45577818451067425325073867435, −10.49780497193256422812101872859, −9.635299070344851436140578045540, −8.465895821087631663891131343601, −7.23320559628268780574252383735, −6.36258030664219151732811234745, −4.93089035679323517899352525674, −4.44638049828875692456451455007, −3.01204329383290617814875961197, −1.74280598325433590533697521094, 1.22759685854150380833617168242, 3.49515449368839939404375506481, 4.66129304708847066975590041053, 5.53486119810001679752324164047, 6.21639806185811834738194993514, 7.25607048063087997277644348944, 8.084819836069008781732669783929, 9.627178711530880398899621304886, 10.16724375800250777461928275994, 11.40491018433566447132389208289

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