Properties

Label 2-429-429.350-c0-0-2
Degree $2$
Conductor $429$
Sign $0.781 - 0.624i$
Analytic cond. $0.214098$
Root an. cond. $0.462708$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 + 1.80i)2-s + (0.809 − 0.587i)3-s + (−2.11 − 1.53i)4-s + (−0.363 − 1.11i)5-s + (0.587 + 1.80i)6-s + (2.48 − 1.80i)8-s + (0.309 − 0.951i)9-s + 2.23·10-s + (0.951 − 0.309i)11-s − 2.61·12-s + (−0.309 + 0.951i)13-s + (−0.951 − 0.690i)15-s + (1.00 + 3.07i)16-s + (1.53 + 1.11i)18-s + (−0.951 + 2.92i)20-s + ⋯
L(s)  = 1  + (−0.587 + 1.80i)2-s + (0.809 − 0.587i)3-s + (−2.11 − 1.53i)4-s + (−0.363 − 1.11i)5-s + (0.587 + 1.80i)6-s + (2.48 − 1.80i)8-s + (0.309 − 0.951i)9-s + 2.23·10-s + (0.951 − 0.309i)11-s − 2.61·12-s + (−0.309 + 0.951i)13-s + (−0.951 − 0.690i)15-s + (1.00 + 3.07i)16-s + (1.53 + 1.11i)18-s + (−0.951 + 2.92i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.781 - 0.624i$
Analytic conductor: \(0.214098\)
Root analytic conductor: \(0.462708\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (350, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :0),\ 0.781 - 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7063709875\)
\(L(\frac12)\) \(\approx\) \(0.7063709875\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−11.65903088490401107567097856052, −9.825425779206595117927571125462, −9.138508017627489683637531519705, −8.599094381886237332205799290960, −7.902473348955992452557548767872, −6.93826688803787273115641422483, −6.25174836855119128107168195073, −4.91491735992342149852661728556, −3.99226320678547777366318435248, −1.29084557054643480212245870162, 2.01248883791131144715996958368, 3.16965587422446130209045635720, 3.67596515694170709200913260421, 4.88767612510308161591205278996, 7.05943520761874582306518574253, 8.097177583582272718559967699311, 8.841662869976046946851248807864, 9.971184197995965109976995318224, 10.16699107324248710804034250246, 11.18616036958467706017399684130

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